The size Ramsey number conjecture for matchings versus multiple paths

Let $n\ge 5$ and $m\ge 3$. For graphs $G$ and $H$, write $\hat{r}(G,H)$ for their size Ramsey number, and let $K_2$ be a single edge and $P_m$ a path on $m$ vertices. The graph $2K_2$ is the matching consisting of two disjoint edges, and $nP_m$ is the disjoint union of $n$ copies of $P_m$. The size Ramsey number conjecture.

$$\hat{r}(2K_2,nP_m)=\min\{nm+1,(n+1)(m-1)\}.$$

The preceding results establish the corresponding formula for small values of $n$, while determining the exact value for general $n$ remains open.