The classification conjecture for graphs attaining local antimagic chromatic number equal to order

Let $G$ be a graph of order $n$, and let $K_n$, $K_{1,n-1}$, $K(m;n_1,n_2,\ldots,n_m)$, and $Ct(3;n_1,0,n_3)$ denote the graph families used in the paper. Order-equality classification conjecture. A graph $G$ of order $n$ has $\chi_{la}(G)=n$ if and only if $G=K_n$ for $n\ge3$, or $G=K_{1,n-1}$ for $n\ge3$, or $G=K(m;n_1,n_2,\ldots,n_m)$ with $(n_m+m-1)(n_m+m)/2>n_1+n_2+\cdots+n_m+\binom{m}{2}$, or $G=Ct(3;n_1,0,n_3)$ with $n_1\le n_2<(n_1+2)(n_1-1)/2$ and $n_1+n_3\equiv0,3\pmod{4}$. The proposed equivalence classifies graphs whose local antimagic chromatic number is as large as possible; the paper gives no resolution of the conjecture.