Saxl's 2-modular tensor-square conjecture

Let $\Bbbk$ be a field of characteristic $2$, let $n=k(k+1)/2$, and let $\rho(k)=(k,k-1,\dots,2,1)$. A partition is 2-regular if no positive part occurs at least twice, and ${\bf D}^\Bbbk(\lambda)$ denotes the corresponding simple $\Bbbk\mathfrak{S}_n$-module. Saxl's 2-modular conjecture. For every 2-regular partition $\lambda$ of $n$,

$$\dim_\Bbbk\left(\operatorname{Hom}_{\mathfrak{S}_n}\left({\bf D}^\Bbbk(\rho(k))\otimes {\bf D}^\Bbbk(\rho(k)),{\bf D}^\Bbbk(\lambda)\right)\right)>0.$$

Equivalently, the tensor square contains all indecomposable projective modules as direct summands with positive multiplicity. This extends the verified complex staircase positivity statement to characteristic $2$; the supplied text gives no resolution status.