Unit-circle products $p_n(z)=\prod_{i\le n}(z-z_i)$: must $\sum_{k\le n}M_k$ exceed $n^{1+c}$? (Erdős #119)

Let $(z_i)_{i\geq 1}$ be an infinite sequence of complex numbers with $\lvert z_i\rvert=1$ for all $i\geq 1$, and for $n\geq 1$ let $$p_n(z)=\prod_{i\leq n} (z-z_i), \qquad M_n=\max_{\lvert z\rvert=1}\lvert p_n(z)\rvert.$$ Erdős asked three questions. (1) Is it true that $\limsup M_n=\infty$? (2) Is it true that there exists $c>0$ such that $M_n>n^c$ for infinitely many $n$? (3) Is it true that there exists $c>0$ such that, for all large $n$, $$\sum_{k\leq n}M_k > n^{1+c}\,?$$ Questions (1) and (2) have been answered affirmatively (see background); the open problem is question (3).

Acceptance. FULLY RESOLVES: a proof that there exists $c>0$ such that EVERY sequence on the unit circle satisfies $\sum_{k\leq n}M_k>n^{1+c}$ for all sufficiently large $n$; OR a disproof — an explicitly defined infinite sequence $(z_i)$ on the unit circle together with a proof that for every $c>0$ the inequality $\sum_{k\leq n}M_k\leq n^{1+c}$ holds for infinitely many $n$ (for instance a sequence with $\sum_{k\leq n}M_k=n^{1+o(1)}$ along a subsequence). Machine-checkable proof preferred (the statement is already formalised in the formal-conjectures Lean repository), else a complete written proof — finite computation alone cannot close this. ADVANCES: an explicit construction with a proven bound $M_n\ll n^{1-c}$ for an exponent strictly better than the best construction stated in the background, or a proven partial-sum bound $\sum_{k\leq n}M_k\ll n^{\theta}$ with $\theta$ strictly smaller than what that construction yields; strengthening Wagner's $(\log n)^c$ or Beck's $N^c$ lower bounds to larger exponents; or a resolution of question (3) restricted to a natural class of sequences (e.g. equidistributed sequences), precisely stated and proved. Deliver the proof or construction manuscript (or Lean file), plus reproducible code and rigorous bound derivations for any computational component.