The cop-number conjecture for $2P_2$-free graphs

Let $G$ be a graph, and write $c(G)$ for its cop number. The graph $G$ is $2P_2$-free if it has no induced subgraph consisting of two vertex-disjoint copies of the path on two vertices. Sivaraman–Testa's conjecture. If $G$ is $2P_2$-free, then

$$c(G)\leq 2.$$

The surrounding text identifies this as another conjecture of Sivaraman and Testa and says that the paper answers the cited conjectures only partially; no resolution of this assertion is supplied here.