Factor-criticality conjecture for K_{1,5}-free 3-vertex-critical graphs

Let $G$ be a finite simple graph. It is factor-critical if, for every vertex $v\in V(G)$, the graph $G-v$ has a perfect matching. The graph $G$ is $K_{1,5}$-free if it has no induced subgraph isomorphic to $K_{1,5}$, and it is 3-vertex-critical if $\gamma(G-v)<\gamma(G)=3$ for every vertex $v\in V(G)$, where $\gamma(G)$ is the domination number. Let $G$ also be 2-connected, have odd order, and satisfy $\delta(G)\geq 3$. Factor-criticality conjecture. Then $G$ is factor-critical. This conjecture concerns when domination-critical graphs necessarily have the matching property that every vertex deletion leaves a perfect matching. Earlier results establish factor-criticality under the stronger assumption that the graph is $K_{1,4}$-free, while the stated $K_{1,5}$-free case is presented as an open conjecture.