The largest-Laplacian-eigenvalue conjecture for Hamming graphs

Let $H(d,q,j)$ be the Hamming graph on $q^d$ vertices, with two vertices adjacent when their Hamming distance is $j$. Write

$$\lambda=q(q-1)^{j-1}{d-1 \choose j-1}.$$

Largest-Laplacian-eigenvalue conjecture. If $q\geq 2$ and $j\geq d-\frac{d-1}{q}$, with $j$ even if $q=2$, then $\lambda$ is the largest Laplacian eigenvalue of $H(d,q,j)$. This conjecture identifies the Laplacian eigenvalue arising from the common Hamming-scheme idempotent as the extremal one in the stated parameter range, extending the known max-cut result for the relevant binary Hamming graphs. Its general status is not established in the supplied source.