The top elementary coefficient 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 with the induced $\mathfrak{S}_n$-action. Write its Frobenius characteristic in the elementary symmetric-function basis:

$$\operatorname{ch}\,\tilde{H}_{n-2}(B_n^{(t)})=\sum_{\lambda\vdash n}a_\lambda e_\lambda.$$

Top elementary coefficient conjecture. The coefficient of $e_n$ is always $1$, that is, $a_{(n)}=1$.

This conjecture concerns the diagonal $\mathfrak{S}_n$-action on the homology of Segre powers of Boolean lattices. The paper reports that Sage computations for $n,t\leqslant7$ support it; its general validity remains open.