Any issue with reaching out directly to interviewer(s) on LinkedIn after rejection when process was handled by a recruiter?

Applied for a position on Indeed. It was listed by a recruiter. They handled coordination with the hiring company to setup interviews and provide feedback.

Company told recruiter they loved me and I made it through the three-round interview process. Ultimately didn't get the job. Recruiter let me know this and also again emphasized that they "loved" me and my skill set, but went with another candidate.

I had already connected with the two people I interviewed with on LinkedIn. It seems there's a lot of talk about if this is good or not, but they work in my industry and are active and even outside of a direct working relationship it makes sense to have that connection.

Is it bad to message them directly to thank them for the opportunity and to let them know I'm still interested if anything else opens up? Should I leave that up to the recruiter completely?

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