The local antimagic chromatic number of generalized book graphs

Let $GB(a_1,a_2,\ldots,a_r)$ denote the generalized book graph obtained by edge-gluing cycles of orders $a_1,a_2,\ldots,a_r$, where $a_1\ge a_2\ge \cdots\ge a_r\ge 3$ and $r\ge 2$. Let $\chi_{la}(G)$ denote the local antimagic chromatic number of a graph $G$. The generalized book graph conjecture. If $a_1\ge 4$, then

$$\chi_{la}(GB(a_1,a_2,\ldots,a_r))=4.$$

The claim concerns the non-triangular generalized book graphs; the paper proves the corresponding value $3$ for $GB(3^{[r]})$ and notes that a $4$-coloring is easy to obtain in the remaining case. The parser supplies no evidence that this statement is resolved, so its status is left open.