The arithmetic-progression variance conjecture for generalized divisor functions

Let $d_z(n)$ be the generalized divisor function and define the arithmetic-progression variance by

$$\mathbb{V}_{d_z}(X,p):=\frac{1}{\phi(p)}\sum_{\substack{1\leq a\leq p\\(a,p)=1}}\left(\sum_{\substack{n\equiv a\bmod p\ \leq X}}d_z(n)-\frac{1}{\phi(p)}\sum_{\substack{(n,p)=1\ \leq X}}d_z(n)\right)^2.$$

Let $a_z$ and $\alpha_1^{(z)}$ be as in the paper. The arithmetic-progression divisor-variance conjecture. For fixed $z>0$ and fixed $\delta\in(0,1)$, as $X\to\infty$ choose primes $p$ with $X/p=X^{\delta+o(1)}$. Then

$$\mathbb{V}_{d_z}(X,p)=a_z\left(\frac{\mathbb{P}(\alpha_1^{(z)}\leq1-\delta)}{\Gamma(z^2)}+o(1)\right)(X/p)(\log X)^{z^2-1}.$$

This is the progression analogue of the divisor-sum short-interval conjecture and remains open.