The $T$-module normalizer conjecture for finite monomial groups

Let $p$ be prime, let $D_{\{p\}'}$ denote the relevant diagonal module, and let $T$ be a non-solvable transitive subgroup of $\mathrm{Sym}(p)$. Write $N_{\mathrm{Sym}(p)}(T)$ for the normalizer of $T$ in $\mathrm{Sym}(p)$. The $T$-module normalizer conjecture. Every finite $T$-submodule of $D_{\{p\}'}$ is an $N_{\mathrm{Sym}(p)}(T)$-submodule. This conjecture concerns the $T$-module listing problem that obstructs the classification of finite irreducible monomial subgroups; the supplied text gives computational evidence and notes that the smallest degree in which it could fail is $73$, but does not state a resolution.