High-moment asymptotic for $L$-functions over small subgroups

Let $k$ be a fixed integer, and let $M_{2k}(p,m)$ denote the $2k$-th moment of the relevant $L$-functions over the subgroup of size $m$. For any $m\mid p-1$, set $d=(p-1)/m$ and suppose that $d$ is odd and

$$\varphi(d)=o(\log p),\qquad\text{as }p\to\infty.$$

Let $\tau_k(n)$ be the $k$-fold divisor function and define

$$a(k)=\sum_{n=1}^{+\infty}\frac{\tau_k^2(n)}{n^2}.$$

High-moment asymptotic conjecture. Under these conditions,

$$M_{2k}(p,m)=a(k)+o(1),\qquad\text{as }p\to\infty.$$

This conjecture extends the proved fourth-moment asymptotic to all fixed even moments and is motivated by the preceding cancellation and fourth-moment results. The expected main term is the Dirichlet series formed from the squared $k$-fold divisor function.