Conjecture on the supremum tail function for Rademacher sums on [√2,2]

Let $\mathcal{X}$ be the class of all Rademacher sums with variance $1$, and define

$$F(x)=\sup_{X\in\mathcal{X}}\mathbb{P}(X>x),\qquad x\in\mathbb{R}.$$

Conjecture for $F$ on $[\sqrt{2},2]$. For $x\in[\sqrt{2},2]$,

$$F(x)=\frac{1}{8}\quad\text{for }x\in[\sqrt{2},\sqrt{3}),$$ $$F(x)=\frac{1}{16}\quad\text{for }x\in[\sqrt{3},2),$$

and

$$F(2)=\frac{9}{256}.$$

The conjecture extends the known result $F(x)=1/4$ for $x\in[1,\sqrt{2})$ to larger values of $x$. Determining $F(x)$ for large $x$ remains beyond current techniques, and the proposed values on this interval are stated as open in the source.