Conrey–Gonek conjecture for long divisor-coefficient Dirichlet polynomials

Let $T$ be sufficiently large and let $K=T^{1+\eta}$ with $\eta\in(0,1)$. For $k\in\mathbb{N}$, define the $k$-th divisor function by

\tau_k(n)=\\#\\{(n_1,\ldots,n_k)\in\mathbb{N}^k:n_1\cdots n_k=n\\}.

Define

$$a_k=\prod_p\left(1-\frac1p\right)^{k^2}\sum_{\alpha=0}^{\infty}\frac{\tau_k^2(p^\alpha)}{p^\alpha},$$ $$w_k(x)=x^{k^2}\left\{1-\sum_{n=0}^{k^2-1}\binom{k^2}{n+1}\gamma_k(n)(-1)^n(1-x^{-n-1})\right\},$$

where

$$\gamma_k(n)=\sum_{i=1}^k\sum_{j=1}^k\binom{k}{i}\binom{k}{j}\binom{n-1}{i+j-2}\binom{i+j-2}{j-1}$$

for $n\in\mathbb{Z}^+$ and $\gamma_k(0)=k$. Conrey–Gonek conjecture. One has

$$\int_T^{2T}\left|\sum_{n\le K}\frac{\tau_k(n)}{n^{1/2+it}}\right|^2\\,dt\sim\frac{a_k}{\Gamma(k^2+1)}w_k\left(\frac{\log K}{\log T}\right)T(\log T)^{k^2}.$$

This conjecture predicts asymptotics for mean values of long Dirichlet polynomials with divisor coefficients and is connected to the sixth and eighth moments of the Riemann zeta function. Its case $k=2$ has been established for every $\eta>0$ by Bettin and Conrey; the general case is therefore solved only in the corresponding established cases.