Here are a couple of problems that I am interested in and are not well-known.
Problem 1 ($100): Prove that . See this paper of mine for definitions and context. As a warm up, one may wish to prove for $25.
Problem 2 ($50): Prove that , where is finite and is rotation by 90 degrees. For a warm–up, $25 for a proof or disproof that . See this paper of mine or problem 5 of Boris Bukh’s website. Update: The warm-up was recently solved by Akshat Mudgal in the affirmative.