Periodicity conjecture for submodule lattices of hook Specht modules

Let $L(r)$ denote the function governing the period in the corresponding result for $2$-part partitions. For integers $n,m$ and fixed $r$, let $\lambda=(n-r,1^r)\vdash n$ and $\mu=(m-r,1^r)\vdash m$, with $n-r>r$ and $m-r>r$. Periodicity conjecture. If

$$n\equiv m\pmod{2^{L(r)}},$$

then the submodule lattices of $S_2^\lambda$ and $S_2^\mu$ are isomorphic. This conjectures that the submodule lattices of hook Specht modules in characteristic $2$ depend periodically on the first-row size, with period controlled by $2^{L(r)}$. The source presents this as an analogue of the periodicity result for $2$-part partitions; no resolution is given here.