For unit-modulus complex numbers $z_i$, let $p_n(z)=\prod_{i\le n}(z-z_i)$ and $M_n=\max_{|z|=1}|p_n(z)|$. Erdős's prize question: is there $c>0$ with $\sum_{k\le n} M_k > n^{1+c}$?

Is it true that $\limsup M_n = \infty$?

This is Problem 4.1 in [Ha74] where it is attributed to Erdős.

The weaker conjecture that $\limsup M_n=\infty$ was proved by Wagner [Wa80], who show that there is some $c>0$ with $M_n>(\log n)^c$ infinitely often.