Asplund–Fox extension conjecture for corona coprime numbers

Let $K_n$ be the complete graph on $n$ vertices, let $\overline{K}_m$ be the edgeless graph on $m$ vertices, and let $\mathfrak{pr}(G)$ denote the minimum coprime number of a graph $G$. Asplund–Fox's corona conjecture. For all positive integers $m$, there exists an $M>m$ such that for all $n>M$,

$$\mathfrak{pr}(K_n\odot\overline{K}_m)=p_{n-1},$$

where $p_{n-1}$ is the $(n-1)$st prime number. The paper's theorem proves the stronger exact formula $\mathfrak{pr}(K_n\odot\overline{K}_m)=\max(mn+n,p_{n-1})$ for all positive integers $m,n$, so this conjecture is solved.