Distributional stability for Khintchine inequalities

Let $G$ be a standard Gaussian random variable, let $X_1,\ldots,X_n$ be independent identically distributed symmetric random variables with

$$\mathbb{E}|X_1|^2=1,$$

and let $a_1,\ldots,a_n$ satisfy the normalization intended in the conjecture. Distributional stability conjecture. For $p\geq 3$, when the variables $X_i$ are sufficiently close in some sense to the Rademacher distribution, one expects, possibly under additional assumptions, that either

$$\mathbb{E}\left|\sum_{i=1}^n a_iX_i\right|^p\leq \mathbb{E}|G|^p$$

or

$$\mathbb{E}\left|\sum_{i=1}^n a_iX_i\right|^p\leq n^{-p/2}\mathbb{E}\left|\sum_{i=1}^n X_i\right|^p$$

holds. The formulation is explicitly tentative: the meaning of closeness and the coefficient assumptions are not specified, and the authors describe the question as largely open.