US Supreme Court Chief Justice John Roberts used both the phrases in conformity with and in conformance with within the same minute during a swearing-in process in the senate. This can be heard in CNN’s video and I am sure found elsewhere.
Answers to “Conformity” vs. “conformance” In English SE draw some distinction between the usage of the two terms.
Justice Roberts being aware of the historic nature of the event and the importance of words will have chosen them carefully.
Does the choice of the two different phrases in two different context suggest somewhat different meanings and implications?
01:35Senators, I attend the Senate in conformity with your notice, for the purpose of joining with you, for the trial of the president of the United States.
02:16At this time I will administer the oath to all senators in the chamber, in conformance with article 1, section 3, clause 6 of the Constitution and the Senate’s impeachment rules.