Optimal fourth-moment deficit in Khintchine's inequality

Let $G$ be a standard Gaussian random variable, let $\varepsilon_1,\ldots,\varepsilon_n$ be independent Rademacher variables, and let $S=\sum_{i=1}^n a_i\varepsilon_i$ satisfy

$$\sum_{i=1}^n a_i^2=1.$$

Optimal fourth-moment deficit conjecture. For every $p\geq 3$,

$$\mathbb{E}|S|^p\leq \mathbb{E}|G|^p-\left(\mathbb{E}|G|^p-1\right)\sum_{i=1}^n a_i^4.$$

The proposed constant agrees with the deficit suggested by the paper's comparison results and is known for even integer $p$; the authors leave the extension to arbitrary real $p\geq 3$ open.