Asplund–Fox conjecture for minimum coprime numbers of joins of paths

Let $P_m$ and $P_n$ be paths on $m$ and $n$ vertices, respectively, and let $\mathfrak{pr}(G)$ denote the minimum coprime number of a graph $G$. Asplund–Fox's path-join conjecture. For any positive integer $N$, there exists a positive integer $M$ such that for all $m>M$ and $2\leq n\leq N$,

\mathfrak{pr}(P_m+P_n)=\begin{cases}m+2n-2&\text{if }m\text{ is odd},\\m+2n-1&\text{if }m\text{ is even}.

The conjecture extends the displayed formula from the cases previously established in the source to every fixed finite range $2\leq n\leq N$ when $m$ is sufficiently large; no resolution is supplied in the provided text.