Kuperberg–Lalin symplectic variance conjecture for quadratic residues modulo primes

Let $k$ be a positive integer and let $p$ be prime. Define

$$\mathcal{S}^S_{d_k;x}(p):=\sum_{\substack{n\leq x\ \equiv\square\,(\mathrm{mod}\,p)\p\nmid n}}d_k(n),$$

where $\square$ denotes a perfect square. Define

$$\operatorname{Var}^w_{p\in[y,2y]}(\mathcal{S}^S_{d_k;x}):=\frac1y\sum_{y<p\leq2y}\log p\left(\mathcal{S}^S_{d_k;x}(p)-\left\langle\mathcal{S}^S_{d_k;x}\right\rangle_S\right)^2,$$

where

$$\left\langle\mathcal{S}^S_{d_k;x}\right\rangle_S:=\frac1{2y}\sum_{y<p\leq2y}\log p\sum_{\substack{n\leq x\p\nmid n}}d_k(n).$$

Kuperberg–Lalin's symplectic prime variance conjecture. For $x^{1/k+\varepsilon}\leq y$, one has

$$\operatorname{Var}^w_{p\in[y,2y]}(\mathcal{S}^S_{d_k;x})\sim a_k^S(\mathcal S)\frac{x}{4}\gamma_{d_k,2}^S\left(\frac{\log x}{\log y}\right)(\log y)^{2k^2+k-2},$$

where

$$a_k^S(\mathcal S)=2\prod_p\left(1-\frac1p\right)^{2k^2+k}\frac12\left(\left(1+\frac1{\sqrt p}\right)^{-2k}+\left(1-\frac1{\sqrt p}\right)^{-2k}\right).$$

The conjecture is derived from a function-field theorem; the source states that it is proved when $y\geq x^2(\log x)^C$ for sufficiently large $C$, while the full stated range remains open.