Perfect-or-bounded-clique-width conjecture for claw-, $4K_1$-, and bridge-free graphs

Let $G$ be a $(claw, 4K_1, bridge)$-free graph.

Perfect-or-bounded-clique-width conjecture. One of the following holds:

• $G$ is perfect.
• $G$ has bounded clique-width.

The conjecture would remove the $C_4$-twin restriction from the paper's polynomial-time colouring result by establishing bounded clique-width whenever such a graph is not perfect. Its status is open in the source.