The divisor variance conjecture for short intervals

For $\alpha>0$ and $k\in\mathbb N$, define $\mathcal C_k(\alpha)$ as the normalized limiting variance of the $k$-fold divisor function in short intervals, and let $\gamma_k(\alpha)$ be the corresponding piecewise-polynomial function. Divisor variance conjecture. For $k\geq1$ and $\alpha>0$,

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

The function $\gamma_k$ is supported on $[0,k]$, symmetric under $\alpha\mapsto k-\alpha$, and piecewise polynomial of degree $k^2-1$. The conjecture is known for $k=2$, with partial results in other ranges, but remains open in general.