Newman–Byrnes merit-factor conjecture for Littlewood polynomials

Let $\mathcal{L}=\bigcup_{N=1}^{+\infty}\{+1,-1\}^N$ be the class of Littlewood sequences, and let $U_N(z)$ denote the associated $L^2$-normalized analytic polynomial for a sequence of length $N$. Newman–Byrnes conjecture.

$$\liminf_{N \longrightarrow +\infty}\left(\min_{u\in\{+1,-1\}^N}\left\|U_N(z)\right\|_4\right)\geq \sqrt[4]{\frac65}.$$

The conjecture is supported in the source by numerical evidence and would solve the merit-factor problem by forcing the minimum $L^4$ norm of $L^2$-normalized analytic polynomials with coefficients $\pm1$ to exceed $1$ by a fixed amount. The source also states that it implies Erdős's conjecture, but gives no resolution status.