Infinite equality cases for the local antimagic chromatic number of lexicographic products

Let $G$ and $H$ be disjoint non-null graphs. Write $G[H]$ for their lexicographic product, and let $\chi_{la}(G)$ denote the local antimagic chromatic number and $\chi(G)$ the chromatic number of $G$. Infinite equality conjecture. There exist infinitely many graphs $G$ and $H$ respectively such that

$$\chi_{la}(G[H])=\chi_{la}(G)\chi_{la}(H)=\chi(G)\chi(H).$$

The paper gives sufficient conditions and examples exhibiting this equality, but the asserted infinitude is left open.