Generalized correlation-immunity bound for perfect 2-colorings of Hamming graphs

Let $H(n,q)$ be the Hamming graph, and let a $(b,c)$-coloring have color parameters $b,c$. Define

$$b'=\frac{b}{\gcd(b,c)},\qquad c'=\frac{c}{\gcd(b,c)}.$$

Generalized correlation-immunity bound. If there exists a $(b,c)$-coloring in $H(n,q)$ with $b'+c'>q$, then

$$n\geq\frac{q+1}{q^2}(b+c).$$

For $q=2$, the corresponding bound is known from correlation immunity; extending it to $q>2$ is presented as an open research problem, and the conjecture begins with $q=3$.