Smallest-eigenvalue conjecture for bilinear forms graphs

Let $B_j(i)$ denote the eigenvalues of the distance-$j$ graph of the bilinear forms association scheme, with parameters $q,d,e$ and $1\le j\le d$. Smallest-eigenvalue conjecture for bilinear forms graphs. For $q\ge3$, or for $q=2$ and $d\ne e$, the smallest eigenvalue in the distance-$j$ graph is $B_j(d-j+1)$ for $1\le j\le d$. The source gives no resolution of this claim, so it remains open.