The Sylow vertex conjecture for hook Specht modules

Let $p$ be an odd prime, let $k$ be a positive integer, and let $r$ be an integer with $0\leq r\leq kp-1$. Write $S^{(kp-r,1^r)}$ for the hook Specht module for the symmetric group $\mathfrak{S}_{kp}$ over a field of characteristic $p$. Sylow vertex conjecture. The module $S^{(kp-r,1^r)}$ has a Sylow $p$-subgroup of $\mathfrak{S}_{kp}$ as a vertex. The conjecture is known in the cases described immediately before it, namely when $r=p$, $k\equiv 1\pmod p$, and $k\not\equiv 1\pmod {p^2}$; its general status is not resolved here.