Rodgers–Soundararajan variance conjecture for the divisor function in arithmetic progressions

Let $k$ be the divisor-function parameter, let $w$ be a smooth function supported in $[1,2]$ satisfying

$$\int w(y)^2\,dy=1,$$

and whose Mellin transform

$$\mathcal{M}[w](s)=\int_0^\infty w(x)x^{s-1}\,dx$$

satisfies rapid decay on every fixed vertical strip. For $X,d\to\infty$ with $\log X/\log d\to c\in(0,k)$, define

$$\Delta_w(\tau_k;X,d,a)=\sum_{n\equiv a\pmod d}\tau_k(n)w\left(\frac nX\right)-\frac1{\varphi(d)}\sum_{(n,d)=1}\tau_k(n)w\left(\frac nX\right),$$

where $\tau_k$ is the $k$-fold divisor function, and define $a_k(d)$ and $\gamma_k(c)$ by

$$a_k(d)=\lim_{s\to1^+}(s-1)^{k^2}\sum_{\substack{n\ge1\\(n,d)=1}}\frac{\tau_k(n)^2}{n^s},$$

with $\gamma_k(c)$ the stated piecewise polynomial involving the Vandermonde determinant and the Barnes $G$-function. Rodgers–Soundararajan's variance conjecture. One has

$$\sum_{\substack{1\le a\le d\\(a,d)=1}}\left|\Delta_w(\tau_k;X,d,a)\right|^2\sim a_k(d)\gamma_k(c)X(\log d)^{k^2-1}.$$

This is a smoothed conjectural asymptotic for the variance of $\tau_k$ in reduced residue classes, motivated by function-field analogues and earlier Barban–Davenport–Halberstam results. The supplied text attributes the formulation to Nguyen and does not provide evidence that it has been resolved.