The equal-extension-count conjecture for finite monomial groups

Let $p$ be prime, let $T$ be a non-solvable transitive subgroup of $\mathrm{Sym}(p)$, and let $D$ be the diagonal subgroup appearing in the monomial group $\widetilde{\mathrm{M}}(p,\mathbb{C})$. A $T$-extension means a subgroup extending a finite $T$-submodule inside $\widetilde{\mathrm{M}}(p,\mathbb{C})$; two extensions are identified when they are conjugate in $\widetilde{\mathrm{M}}(p,\mathbb{C})$. The equal-extension-count conjecture. Every finite $T$-submodule of $D$ has the same number of $T$-extensions in $\widetilde{\mathrm{M}}(p,\mathbb{C})$ up to $\widetilde{\mathrm{M}}(p,\mathbb{C})$-conjugacy. This conjecture addresses the extension problem for projective $T$ in the classification of finite irreducible monomial subgroups. The source reports verification in several cases and gives the corresponding extension counts, but does not state a general resolution.