Smooth-cutoff variance conjecture for the k-fold divisor function in arithmetic progressions

Fix $k\ge2$. Let $w(y)$ be a smooth function supported in $[1,2]$ such that

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

and whose Mellin transform $\mathcal{M}[w]$ satisfies

$$\mathcal{M}[w](\sigma+it)\ll_\ell\frac{1}{1+|t|^\ell}$$

uniformly for $|\sigma|\le A$, for every fixed $A>0$ and every positive integer $\ell$. Let $\Delta_w(\tau_k;X,d,a)$ denote the smoothed divisor-function error term in an arithmetic progression, and let $a_k(d)$ and $\gamma_k(c)$ be the arithmetic constant and piecewise polynomial defined above. Smooth-cutoff variance conjecture. If $X,d\to\infty$ with $\log X/\log d\to c\in(0,k)$, then

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

This conjecture extends the known smoothed variance asymptotic in the range handled by the paper to all $c\in(0,k)$. The source indicates that the formula is known in the range $0<c<1$ through earlier work, while the full range remains open; endpoint and off-diagonal difficulties obstruct the current method.