Uniform Berry–Esseen conjecture for Rademacher sums

Let $x_1,\ldots,x_n$ be independent random variables uniformly distributed in $\{-1,1\}$, and let

$$X=\sum_i a_i x_i$$

be a Rademacher sum with $\operatorname{Var}(X)=1$ and $0<a_i\leq a_1$ for every $i$. Let $Z\sim N(0,1)$ be a standard Gaussian, and let $x\in\mathbb R$. Berry–Esseen conjecture.

$$\left|\Pr[X\leq x]-\Pr[Z\leq x]\right|\leq \Pr[Z\in(0,a_1)]<\frac{a_1}{\sqrt{2\pi}}.$$

This is proposed as a natural extension of a previously proved bound for coefficients bounded by $0.22$. The source presents it as an open problem, with no resolution supplied.