Bilinear forms graph smallest eigenvalue conjecture

Let $H_q(d,e)$ be the bilinear forms scheme on the $d\times e$ matrices over $\mathbb{F}_q$, where $d\leq e$. In its distance-$j$ graph, write $B_j(i)$ for the eigenvalue indexed by $i$, with $0\leq i,j\leq d$.

Bilinear forms eigenvalue conjecture. For $q\geq 3$, or $q=2$ and $d\neq e$, $B_j(d-j+1)$ is the smallest eigenvalue in the distance-$j$ graph for $1\leq j\leq d$.

The conjecture was previously proved for $q\geq 4$ and is proved in this paper for all $q\geq 2$, so the claim is now resolved.