Exact threshold conjecture for adding three-paths

Let $G$ be a graph and let $t$ be a positive integer. Write $G\cup tP_3$ for the disjoint union of $G$ and $t$ copies of $P_3$, and let $\tau(G)$ be the maximum integer such that $G\cup sP_3$ is antimagic for every integer $s\leqslant\tau(G)$. Exact threshold conjecture. $G\cup tP_3$ is antimagic if and only if $t\leqslant\tau(G)$. Equivalently, $\tau(G)$ is the maximum integer $t$ such that $G\cup tP_3$ is antimagic. The source presents this as a conjectural strengthening of its upper bounds; it would settle the remaining cases in which the established bound $\beta(G)$ may exceed $\tau(G)$.