The complete-join characterization of local antimagic chromatic number

Let $G$ be a graph, let $K_n$ be the complete graph on $n$ vertices, and let $G\vee K_n$ denote their join. Write $\chi(G)$ for the chromatic number and $\chi_{la}(G)$ for the local antimagic chromatic number.

Complete-join characterization. For $n\ge 2$,

$$\chi_{la}(G\vee K_n)\ge\chi_{la}(G)+n$$

if and only if

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

The conjecture proposes an exact criterion for the local antimagic chromatic number to increase by at least $n$ under joining with a complete graph. The supplied text states the claim but gives no resolution, so its status remains open.