Characterization of equality for local antimagic chromatic numbers of lexicographic products

Let $G$ and $H$ be graphs, let $G[H]$ be their lexicographic product, and let $\chi_{la}$ and $\chi$ denote the local antimagic chromatic number and chromatic number, respectively. Suppose that $k\geq 1$ and that $2k+1$ is the length of a shortest odd cycle in $G$. Equality characterization conjecture. One has

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

if and only if

$$\chi(G)\chi(H)=2\chi(H)+\left\lceil\frac{\chi(H)}{k}\right\rceil.$$

The paper develops upper bounds and examples for local antimagic chromatic numbers of lexicographic products; this proposed characterization remains open.