Product lower-bound question for complete graph immersions

Determine whether, for arbitrary graphs G and H with im(G)=t and im(H)=r, and for each of the four standard graph products \ast, the inequality im(G\ast H)\geq im(K_t\ast K_r) holds.

Background

Here im(G) is the order of the largest complete graph immersion in G. Collins, Heenehan, and McDonald posed this question to determine whether the largest complete immersion in a product is bounded below by the corresponding product of complete graphs having the same immersion numbers.

The paper states that the question has been answered positively for lexicographic and Cartesian products, while the direct and strong product cases were conjectured separately and remained unresolved in the cited discussion.

References

Letting $im(G)$ be the order of the largest complete graph immersion of $G$, called the \underline{immersion number} of $G$, they raised the following question. Let $G$ and $H$ be graphs with $im(G) = t$ and $im(H) = r$, and let $\ast$ be any of the four standard graph products. Is $im(G \ast H) \geq im(K_t \ast K_r)$?

— Totally odd immersions of complete graphs in graph products  (2502.10227 - Echeverría et al., 14 Feb 2025) in Section 1, Introduction; Question 1 (labelled \ref{qus:im})