Polylogarithmic gap conjecture for sums of two squares

Let $\mathcal S=\{s_1<s_2<\ldots<s_n<\ldots\}$ be the set of natural numbers expressible as the sum of two squares of integers, and define

$$R(x)=\min_{n\in\mathcal S}|x-n|.$$

Gap conjecture for sums of two squares. For every $\varepsilon>0$,

$$R(x)\ll_{\varepsilon}x^{\varepsilon}.$$

The known bound is $R(x)\ll x^{1/4}$, while it remains unknown whether even $R(x)=o(x^{1/4})$ holds. The conjecture predicts that the gaps near $x$ are smaller than every fixed positive power of $x$.