Grassmann graph eigenvalue monotonicity and exceptional minimum conjecture

Let $V$ be a vector space of dimension $n$ over $\mathbb{F}_q$, with $n\rect 2d$. The vertices of the Grassmann scheme $G_q(n,d)$ are the $d$-dimensional subspaces of $V$, and $G_j(i)$ denotes the eigenvalue of the distance-$j$ graph indexed by $i$, for $0\rect i,j\rect d$.

Grassmann eigenvalue conjecture. (i) If $(n,q) eq (2d,2)$, then

$$|G_j(i+1)| < |G_j(i)|$$

where $0\rect i\rect d-1$. (ii) If $(n,q)=(2d,2)$, then $G_j(d-j)$ is negative for $(d,j)=(5,3)$ and when $d\rect 6$, $2\rect j\rect d-2$, and $G_j(d-j)$ is the smallest among the $G_j(i)$ when $d\rect 6$, $3\rect j\rect d-2$.

Part (i) is proved for $q\rect 3$ in the paper, while part (ii) is known for $7\rect j\rect d-5$ and remains open for the listed exceptional values of $j$.