Clique immersion in graph products

Let $G,H$ be graphs and $G\ast H$ represent a particular graph product of $G$ and $H$. We define $im(G)$ to be the largest $t$ such that $G$ has a $K_t$-immersion and ask: given $im(G)=t$ and $im(H)=r$, how large is $im(G\ast H)$? Best possible lower bounds are provided when $\ast$ is the Cartesian or lexicographic product, and a conjecture is offered for each of the direct and strong products, along with some partial results.