Stability conjecture for the third Khintchine moment

Let $S=\sum_{i=1}^n a_i\varepsilon_i$ be a Rademacher sum, let $S_n=n^{-1/2}\sum_{i=1}^n\varepsilon_i$, and suppose that

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

Third-moment stability conjecture. There exists a universal constant $C_3>0$ such that

$$\mathbb{E}|S|^3\leq \mathbb{E}|S_n|^3-C_3\sum_{i=1}^n\left(a_i^2-\frac{1}{n}\right)^2.$$

This conjectures quantitative stability of the optimal third-moment Khintchine inequality, with equality attained by the equal-coefficient sum. The paper notes that the currently obtained constant can likely be improved, but leaves this stability statement open.