Given a weight $w$ on the unit circle, consider the orthogonal polynomials on the unit circle generated by $w$. Steklov famously conjectured that if $w$ is bounded below, then the polynomials all ought to be uniformly bounded above. While false, this conjecture begs the follow-up question: under what regularity conditions on $w$ are the polynomials uniformly bounded in $L^p(w)$ for some $p\gt 2$? Building upon a preliminary answer given by Nazarov for when $w$ is bounded above and below, we provide a positive answer when $w$ is an $A_2$ weight. This is joint work with Alexander Aptekarev and Sergey Denisov.