The maximal nilpotency conjecture for $\mathsf{Ver}_{p^\infty}$

Let $\mathcal{C}$ be a tensor category. For every simple object $L\ne\mathbbm{1}$, require that $\operatorname{Sym} L$ be finite, and for every non-split monomorphism $\alpha:\mathbbm{1}\hookrightarrow X$, require that $\alpha^n:\mathbbm{1}\to\operatorname{Sym}^nX$ vanish for some $n$. A tensor category satisfying both conditions is maximally nilpotent, or MN. Maximal nilpotency conjecture. The category $\mathsf{Ver}_{p^\infty}$ is MN. Maximal nilpotency is characterized in the surrounding text by nilpotence of the radical in every algebra object and by the integral-domain property of domains; the source gives no resolution status for this assertion.