The divisor variance conjecture in arithmetic progressions

Let $q$ tend to infinity through primes, let $\mathcal D_k(\alpha)$ be the normalized variance of the $k$-fold divisor function in reduced residue classes, and let $\gamma_k(\alpha)$ be the same piecewise-polynomial function as in the short-interval problem. Arithmetic-progression divisor variance conjecture. For $k\geq1$ and $\alpha>0$,

$$\mathcal D_k(\alpha)=\gamma_k(\alpha).$$

The conjecture is known for $k=2$ and has partial results in other ranges; smoothed averaged versions are known in restricted ranges, while the full pointwise statement remains open.