Upper domination conjecture for direct products of complete multipartite graphs

Let $K[a_i,b_i]$ denote the complete $b_i$-partite graph with $a_i$ vertices in each part, and let $G$ be its direct product

$$G= \prod_{i=1}^t K[a_i,b_i].$$

Assume $2\leq b_1\leq b_2\leq\cdots\leq b_t$. The upper domination conjecture.

$$\Gamma(G)=\frac{1}{b_1}\prod_{i=1}^t a_i b_i.$$

Here $\Gamma(G)$ is the maximum size of a minimal dominating set of $G$. The proposition preceding the conjecture proves the corresponding lower bound, and the equality is known when $t\leq 3$ but remains open in general. A particularly attractive unresolved special case is $G=\prod_{i=1}^t K_3$.