Joshi–Petrov conjecture on nonsmooth Hecke algebras of Drinfeld modular forms

Let $q$ be a prime power, let $K={\mathbb F}_q(T)$, and let $\mathcal{H}_{k,m}$ be the Hecke algebra acting on the space $M_{k,m}$ of Drinfeld modular forms of weight $k$ and type $m$. Here $0\leq m<q$ and $k\equiv 2m\pmod{q-1}$.

Joshi–Petrov conjecture. Given $q$, there exist $k\gg 0$ and $m$, with $0\leq m<q$ and $k\equiv 2m\pmod{q-1}$, such that

$$H^2(\mathcal{H}_{k,m},M_{k,m})\neq 0.$$

The conjecture predicts that the natural Hecke module of Drinfeld modular forms detects nonsmoothness of the Hecke algebra for sufficiently large weights. It is stronger than the assertion that the Hecke algebra is nonsmooth, since that only requires nonvanishing cohomology for some module.