Linear-threshold conjecture for connected size Ramsey numbers of star matchings

Let $nK_{1,p}$ be the disjoint union of $n$ copies of the star $K_{1,p}$, and let $K_{1,m}$ be the star with $m$ edges. The connected size Ramsey number $\hat r_c(G_1,G_2)$ is the minimum number of edges in a connected graph $G$ such that $G\to (G_1,G_2)$. Linear-threshold conjecture. There exists a positive constant $c$ such that for all positive integers $m$, $n$, and $p$ with $m\geq c(n+p)$, we have

$$\hat r_c(nK_{1,p},K_{1,m})=n(m+p)-1.$$

The conjecture proposes that the exact value established in the paper for the stronger condition $m\geq (n^2+2pn+n-3)/2$ remains valid once $m$ is merely linear in $n+p$.