Dichotomy conjecture for finite families of vertex-critical H-free graphs

Let $k\geq 5$, let $H$ be a graph, and let a graph be $k$-vertex-critical if it has chromatic number $k$ and deleting any vertex lowers its chromatic number. A graph is $H$-free if it has no induced subgraph isomorphic to $H$.

Dichotomy conjecture. There is a finite number of $k$-vertex-critical $H$-free graphs if and only if $H$ is an induced subgraph of $P_4+sP_1$ for some $s\geq 0$.

This conjecture would resolve the open problems concerning the finiteness of $k$-vertex-critical graphs avoiding $P_3+2P_1$, $P_4+P_1$, and their generalizations to graphs on at least five vertices. The supplied text does not state whether the conjecture has been resolved.