Symplectic variance conjecture for sums of divisor functions over quadratic residues

Let $d_k(n)$ denote the $k$-fold divisor function, and define

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

For $x^{1/k}\leq y$ and primes $p$ with $y\leq p\leq 2y$, write $\mathrm{Var}_{p\in[y,2y]}$ for the variance over these primes. The symplectic variance conjecture.

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

where $a_k^S$ is an arithmetic constant and $\gamma_{d_k,2}^S(c)$ is the piecewise polynomial of degree $2k^2+k-2$ given by the source. This conjecture predicts the integer analogue of the symplectic variance arising from the corresponding function-field integral; its status is not resolved in the supplied text.