Goedgebeur and Schaudt's conjecture on 4-vertex-critical $\{P_7,C_3\}$-free graphs

Let $P_7$ denote the path on seven vertices and let $C_3$ denote the cycle on three vertices. A graph is 4-vertex-critical if its chromatic number is $4$ and deleting any vertex lowers its chromatic number. A graph is $\{P_7,C_3\}$-free if it has no induced subgraph isomorphic to $P_7$ or $C_3$. Goedgebeur–Schaudt's conjecture. There are exactly seven 4-vertex-critical $\{P_7,C_3\}$-free graphs. The conjecture refines the known finiteness result for this class, which gives an upper bound on the order of such graphs but does not determine the exact list.