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Let $G = (V,E)$ be a connected simple graph. A bijection $f: E \rightarrow\{1,2,\ldots,|E|\}$ is called a local antimagic labeling of $G$ if $f^+(u) \neqf^+(v)$ holds for any two adjacent vertices $u$ and $v$, where $f^+(u) =\sum_{e\in E(u)} f(e)$ and $E(u$) is the set of edges incident to $u$. A graph$G$ is called local antimagic if $G$ admits at least a local antimagiclabeling. The local antimagic chromatic number, denoted $\chi_{la}(G)$, is theminimum number of induced colors taken over local antimagic labelings of $G$.Let $G$ and $H$ be two disjoint graphs. The graph $G[H]$ is obtained by thelexicographic product of $G$ and $H$. In this paper, we obtain sufficientconditions for $\chi_{la}(G[H])\leq \chi_{la}(G)\chi_{la}(H)$. Consequently, wegive examples of $G$ and $H$ such that $\chi_{la}(G[H]) = \chi(G)\chi(H)$,where $\chi(G)$ is the chromatic number of $G$. We conjecture that (i) thereare infinitely many graphs $G$ and $H$ such that$\chi_{la}(G[H])=\chi_{la}(G)\chi_{la}(H) = \chi(G)\chi(H)$, and (ii) for $k\ge1$, $\chi_{la}(G[H]) = \chi(G)\chi(H)$ if and only if $\chi(G)\chi(H) =2\chi(H) + \lceil\frac{\chi(H)}{k}\rceil$, where $2k+1$ is the length of ashortest odd cycle in $G$.

A note on local antimagic chromatic number of lexicographic product graphs