Kuperberg–Lalin symplectic variance conjecture for quadratic character sums

Let $k$ be a positive integer, let $r$ be a positive fundamental discriminant, and let $\chi_r$ be the primitive quadratic character modulo $r$. Define

$$\mathcal{T}^S_{d_k;x}(r):=\sum_{\substack{n\leq x\\(n,r)=1}}d_k(n)\chi_r(n).$$

For $y>0$, define the variance over positive fundamental discriminants in $(y,2y]$ by

$$\operatorname{Var}_{r\in(y,2y]}(\mathcal{T}^S_{d_k;x}):=\mathbb E^*_{y<r\leq 2y}\bigl(\mathcal{T}^S_{d_k;x}(r)\bigr)^2.$$

Here $\mathbb E^*$ denotes expectation over positive fundamental discriminants. Kuperberg–Lalin's symplectic variance conjecture. If $x^{1/k+\varepsilon}\leq y$, then

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

where

$$a_k^S(\mathcal T)=2\prod_p\left(1-\frac1p\right)^{k(2k+1)}\left(\frac1{p+1}\left(1+\frac p2\left(\left(1+\frac1{\sqrt p}\right)^{-2k}+\left(1-\frac1{\sqrt p}\right)^{-2k}\right)\right)\right),$$

and $\gamma_{d_k,2}^S(c)$ is the piecewise polynomial of degree $2k^2+k-2$ defined as the leading coefficient in $N$ of

$$I_{d_k,2}^S(c2N;N):=\int_{\operatorname{Sp}(2N)}\left|\sum_{\substack{j_1+\cdots+j_k=c2N\\0\leq j_1,\ldots,j_k\leq 2N}}\operatorname{Sc}_{j_1}(U)\cdots\operatorname{Sc}_{j_k}(U)\right|^2\,\mathrm dU.$$

The conjecture is motivated by an analogous function-field result and incorporates the arithmetic factor $a_k^S(\mathcal T)$. The source states that it is proved when $y\geq x^2(\log x)^C$ for a sufficiently large constant $C$, leaving the stated range beyond that regime open.