Hermitian forms graph smallest eigenvalue conjectures

Let $Q_q(d)$ be the Hermitian forms scheme on the $d\times d$ Hermitian matrices over $\mathbb{F}_{q^2}$, and let $Q_j(i)$ denote the eigenvalue indexed by $i$ in its distance-$j$ graph, for $0\leq i,j\leq d$.

Hermitian forms eigenvalue conjectures. (i) If $j$ is odd, then

$$Q_j(1) \leq Q_j(i)$$

for $0\leq i\leq d$. (ii) If $j$ is even and $j\geq 2$, then

$$Q_j(d-j+2) \leq Q_j(i)$$

for $0\leq i\leq d$.

Both assertions are proved in the paper for $q\geq 2$, $d\geq 6$, and $j\geq 1$.