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>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 A2 weight. This is joint work with Alexander Aptekarev and Sergey Denisov.