Direct and strong product conjectures for immersion number

Prove that the inequality im(G\ast H)\geq im(K_t\ast K_r) holds for the direct product and the strong product of arbitrary graphs G and H, where im(G)=t and im(H)=r.

Background

The cited work established the product lower bound for lexicographic and Cartesian products. The paper explicitly reports that Collins, Heenehan, and McDonald conjectured that the same positive answer should hold for the direct (tensor) and strong products.

The present paper proves an analogous inequality for the parameter toi measuring totally odd strong immersions, not the ordinary immersion number im. Thus the ordinary immersion-number conjectures remain unresolved in the passage quoted.