Fourth-moment conjectures for Rademacher sums

Let $\varepsilon_1,\ldots,\varepsilon_n$ be independent Rademacher signs, let $\sum a_i^2 = 1$, write $S = \sum a_i\varepsilon_i$ and $q = \sum a_i^4$, and let $\mu_p = \mathbb{E}|G|^p$ for a standard Gaussian $G$. Two 2025 conjectures say that $q$ alone governs how far $S$ falls short of Gaussian. Jakimiuk proved $\mathbb{E}|S|^p \le \mu_p - c_p q$ for $p \ge 3$ and conjectured the optimal constant is $c_p = \mu_p - 1$ throughout that range; separately he conjectured a dimension-free quadratic stability bound at the critical exponent $p = 3$. Baranski, Murawski, Nayar and Oleszkiewicz reduced the finite-dimensional $L_p/L_4$ Khintchine constant for $p \ge 5$ to $\sup_{x \ge 1}\|x + \varepsilon_1 + \cdots + \varepsilon_N\|_p / \|x + \varepsilon_1 + \cdots + \varepsilon_N\|_4$ and conjectured the supremum is attained at $x = 1$ - that is, the flat coefficient vector is the extremizer.

Four results, and the first is partly a refutation. Jakimiuk conjectured $c_p = \mu_p - 1$ is optimal for every $p \ge 3$; the paper proves that for $p \ge 4$ and gives a counterexample for every $2 < p < 4$, so the conjecture is false as posed and the corrected range is $p \ge 4$. The witness is the two-coordinate vector $S_2 = (\varepsilon_1+\varepsilon_2)/\sqrt2$. The Baranski-Murawski-Nayar-Oleszkiewicz flat-point conjecture is proved outright, in the stronger form that $x \mapsto \|x+S_n\|_p/\|x+S_n\|_4$ is strictly decreasing on $[1,\infty)$ for every real $p \ge 5$; that range is the one they conjectured, so nothing is left over. Jakimiuk's second conjecture, dimension-free quadratic stability at $p = 3$, is proved with an explicit constant, though the optimal constant there is only bracketed and stays open. The paper also records the exact fixed-$q$ moment and Laplace-transform envelopes, from which coefficient-sensitive tail bounds follow.