Hamming graph distinct-eigenvalue conjecture

Let $H(d,q)$ be the Hamming scheme on $Q^d$, where $Q$ is a set of size $q$, and let $H(d,q,j)$ be its distance-$j$ graph.

Hamming distinct-eigenvalue conjecture. If $H(d,q,j)$ is connected, it has more than $d/2$ distinct eigenvalues.

The paper explicitly states that this conjecture remains open.