The uniseriality conjecture for indecomposable $T$-submodules

Let $p$ be prime, let $T$ be a non-solvable transitive subgroup of $\mathrm{Sym}(p)$, and let $D$ be the diagonal module used in the monomial-group classification. A module is uniserial if its submodules form a chain. The uniseriality conjecture. Every indecomposable $T$-submodule of $D$ is uniserial. This is another proposed ingredient for resolving the module-listing and extension obstacles in the classification of finite irreducible monomial subgroups. The source describes supporting patterns and case-by-case evidence, but gives no general proof or disproof.