The sharp lower-bound conjecture for multicolor Hamming graphs

Let $H(n,q)$ denote the $q$-ary Hamming graph, and let $b(H(n,q))$ denote its bipartite independence-related parameter as defined in the paper. Fix an integer $q\geq 3$. Sharp-bound conjecture. For sufficiently large $n$,

$$b(H(n,q))=\left\lfloor\left(1-\tfrac1q\right)n+\tfrac{q+1}2\right\rfloor.$$

The surrounding discussion says that the author could not extend Alon's argument to $H(n,q)$ for $q\geq 3$, suggesting that the lower bound in the main theorem may be improvable. Thus the proposed equality is not established in the supplied text.