Let $P$ be a convex polygon. Suppose every interior angle of $P$ is obtuse. Is it always the case that there exist three vertices $p, q, r$ of $P$ such that $\triangle pqr$ is acute?

I conjecture that the answer is yes. I have tried different types of triangulations of $P$ (e.g. fan triangulation, triangulation by repeatedly connecting every other vertex, etc.) However, I haven't come up with a proof yet.

Edit: And how about if $P$ has at most one acute angle (others stay obtuse)? I think the result should be the same.