Beginning on page 23 of the Court's ruling, the majority writes (emphasis added):

When the Vice President presides over the January 6 certification proceeding, he does so in his capacity as President of the Senate. Ibid. Despite the Vice President’s expansive role of advising and assisting the President within the Executive Branch, the Vice President’s Article I responsibility of “presiding over the Senate” is “not an ‘executive branch’ function.” [...] With respect to the certification proceeding in particular, Congress has legislated extensively to define the Vice President’s role in the counting of the electoral votes, see, e.g., 3 U. S. C. §15, and the President plays no direct constitutional or statutory role in that process. So the Government may argue that consideration of the President’s communications with the Vice President concerning the certification proceeding does not pose “dangers of intrusion on the authority and functions of the Executive Branch.”

The example provided is of communication between the President and Vice President regarding the latter's role in certifying the results of the presidential election. The majority says that since the President has no constitutional role in that certification, such discussions are arguably fair game for prosecutors. However, the court doesn't definitively propose any particular set of facts which would be sufficient for the government to establish that and for the Court to agree.

I did not find any other examples in the majority's opinion of what circumstances might be sufficient to overcome a claim of presumptive immunity.

No.

The closest it comes to doing so begins on page 23 of the Court's ruling, where the majority writes (emphasis added):

When the Vice President presides over the January 6 certification proceeding, he does so in his capacity as President of the Senate. Ibid. Despite the Vice President’s expansive role of advising and assisting the President within the Executive Branch, the Vice President’s Article I responsibility of “presiding over the Senate” is “not an ‘executive branch’ function.” [...] With respect to the certification proceeding in particular, Congress has legislated extensively to define the Vice President’s role in the counting of the electoral votes, see, e.g., 3 U.S.C. §15, and the President plays no direct constitutional or statutory role in that process. So the Government may argue that consideration of the President’s communications with the Vice President concerning the certification proceeding does not pose “dangers of intrusion on the authority and functions of the Executive Branch.”

The example provided is of communication between the President and Vice President regarding the latter's role in certifying the results of the presidential election. The majority says that since the President has no constitutional role in that certification, such discussions are arguably fair game for prosecutors. However, the court doesn't definitively propose any particular set of facts which would be sufficient for the government to establish that and for the Court to agree.

I did not find any other examples in the majority's opinion of what circumstances might be sufficient to overcome a claim of presumptive immunity.

Beginning on page 23 of the court's ruling, the majority writes (emphasis added):

When the Vice President presides over the January 6 certification proceeding, he does so in his capacity as President of the Senate. Ibid. Despite the Vice President’s expansive role of advising and assisting the President within the Executive Branch, the Vice President’s Article I responsibility of “presiding over the Senate” is “not an ‘executive branch’ function.” [...] With respect to the certification proceeding in particular, Congress has legislated extensively to define the Vice President’s role in the counting of the electoral votes, see, e.g., 3 U. S. C. §15, and the President plays no direct constitutional or statutory role in that process. So the Government may argue that consideration of the President’s communications with the Vice President concerning the certification proceeding does not pose “dangers of intrusion on the authority and functions of the Executive Branch.”

The example provided is of communication between the President and Vice President regarding the latter's role in certifying the results of the presidential election. The majority says that since the President has no constitutional role in that certification, such discussions are arguably fair game for prosecutors. However, the court doesn't definitively propose any particular set of facts which would be sufficient for the government to establish that and for the court to agree.

I did not find any other examples in the majority's opinion of what circumstances might be sufficient to overcome a claim of presumptive immunity.

Beginning on page 23 of the Court's ruling, the majority writes (emphasis added):

When the Vice President presides over the January 6 certification proceeding, he does so in his capacity as President of the Senate. Ibid. Despite the Vice President’s expansive role of advising and assisting the President within the Executive Branch, the Vice President’s Article I responsibility of “presiding over the Senate” is “not an ‘executive branch’ function.” [...] With respect to the certification proceeding in particular, Congress has legislated extensively to define the Vice President’s role in the counting of the electoral votes, see, e.g., 3 U. S. C. §15, and the President plays no direct constitutional or statutory role in that process. So the Government may argue that consideration of the President’s communications with the Vice President concerning the certification proceeding does not pose “dangers of intrusion on the authority and functions of the Executive Branch.”

The example provided is of communication between the President and Vice President regarding the latter's role in certifying the results of the presidential election. The majority says that since the President has no constitutional role in that certification, such discussions are arguably fair game for prosecutors. However, the court doesn't definitively propose any particular set of facts which would be sufficient for the government to establish that and for the Court to agree.

I did not find any other examples in the majority's opinion of what circumstances might be sufficient to overcome a claim of presumptive immunity.