Immersion-number product inequality for direct and strong products

Establish that for graphs G and H with immersion numbers im(G)=t and im(H)=r, the inequality im(G * H) ≥ im(K_t * K_r) holds when * is respectively the direct product or the strong product.

Background

Collins, Heenehan, and McDonald asked whether the largest complete-graph immersion in a product G * H is always at least as large as the corresponding quantity in the product of complete graphs determined by im(G) and im(H). The paper states that the inequality was proved for lexicographic and Cartesian products, while a positive answer for the direct and strong products remained conjectural. The present paper addresses the analogous parameter for totally odd strong immersions rather than resolving this ordinary immersion-number conjecture.