Non-distributivity conjecture for hook Specht modules in characteristic 2

Let $\lambda=(n-r,1^r)\vdash n$ with $0\leq r\leq n-r$. A hook Specht module $S_2^\lambda$ is called distributive when its submodule lattice is a distributive lattice, equivalently when

$$U\cap(V+W)=(U\cap V)+(U\cap W)$$

for all submodules $U,V,W$ of $S_2^\lambda$.

Non-distributivity conjecture. If $r\geq 10$, then $S_2^\lambda$ is not distributive.

This conjecture asks for a combinatorial understanding of when hook Specht modules in characteristic $2$ have distributive submodule lattices. The paper establishes that Specht modules indexed by 2-part partitions are distributive, while the asserted non-distributivity for hooks with $r\geq 10$ remains open.