Direct-product immersion conjecture

Let $G$ and $H$ be graphs, and write $\operatorname{im}(G)=t$ and $\operatorname{im}(H)=r$. Direct-product immersion conjecture. Then

$$\operatorname{im}(G\times H)\geq (t-1)(r-1)+1.$$

This conjecture asserts that the direct product preserves the proposed lower bound for immersion numbers; the paper gives partial results but does not resolve the general case.