Van Dam–Sotirov's smallest-eigenvalue conjecture for Hamming graphs

Let $q \ge 2$ and $d \ge 1$ be integers, let $Q$ be a set of size $q$, and let $H(d,q,j)$ be the distance-$j$ graph of the Hamming scheme on $Q^d$, whose eigenvalues are $K_j(i)$, where

$$K_j(i)=\sum_{h=0}^j(-1)^h(q-1)^{j-h}\binom{i}{h}\binom{d-i}{j-h}.$$

Van Dam–Sotirov's conjecture. If $j\ge d-\frac{d-1}{q}$, with $j$ even when $q=2$, then the smallest eigenvalue of $H(d,q,j)$ is $K_j(1)$. The conjecture was proved in the paper: the binary case was already known, and the nonbinary case is settled here, so it is solved.