A thank-you note is optional. It is rarely what determines whether you move forward after an MBB interview.

The interview evaluation is usually based on the interview itself, so do not treat the note as a chance to repair a weak case or add a second recommendation after the fact.

If you had a particularly useful conversation, a short message thanking the interviewer for their time is completely reasonable. Two or three sentences is enough.

Do not chase down personal contact information if the firm has not given it to you. If communication has been routed through recruiting, it is also fine to simply thank the recruiter and leave the formal evaluation process alone.

I sent one after R2 because the conversation was genuinely helpful, but skipped it after R1.