Irreducible-support stability conjecture for Segre powers of Boolean lattices

Let $B_n^{(t)}$ denote the $t$-fold Segre power of the Boolean lattice, and let $\tilde{H}_{n-2}(B_n^{(t)})$ be its reduced homology representation of $\mathfrak{S}_n$. An irreducible $\mathfrak{S}_n$-representation appears in $\tilde{H}_{n-2}(B_n^{(t)})$ when it occurs as a constituent of that representation.

Irreducible-support stability conjecture. For $t\geqslant3$, every irreducible appearing in $\tilde{H}_{n-2}(B_n^{(t)})$ also appears in $\tilde{H}_{n-2}(B_n^{(2)})$.

This is proposed as a stability statement for the set of irreducible constituents in the homology of Segre powers. The source reports computational support for the surrounding diagonal-action conjectures for $n,t\leqslant7$, but does not state that this conjecture has been proved or refuted.