Conjecture for minimum coprime numbers of joins of paths

Let $P_m$ and $P_n$ be paths on $m+1$ and $n+1$ vertices, respectively, and let $P_m+P_n$ denote their graph join. The minimum coprime number $\mathfrak{pr}(G)$ is the least $M$ for which the vertices of $G$ have distinct labels from $1,\ldots,M$ such that adjacent labels are relatively prime. The path-join conjecture. For positive integers $m,n$ with $m\geq n$,

$$\mathfrak{pr}(P_m+P_n)=m+2n-2+\frac{1-(-1)^m}{2}.$$

This strengthens the preceding bounded-parameter conjectural upper bound and would determine the minimum coprime number for every join of two paths. The source does not provide a resolution.