The narrowness conjecture for $H$-free graphs

Let $G$ be a graph. A function $f:V(G)\to[0,1]$ is good if, for every perfect induced subgraph $P$ of $G$,

$$\sum_{v\in V(P)} f(v)\leq 1.$$

For $\alpha\geq 1$, the graph $G$ is $\alpha$-narrow if, for every good function $f$,

$$\sum_{v\in V(G)} f(v)^\alpha\leq 1.$$

Narrowness conjecture. For every graph $H$, there exists a constant $\alpha(H)\geq 1$ such that every $H$-free graph $G$ is $\alpha(H)$-narrow.

Fox showed that this conjecture is equivalent to the Erdős–Hajnal conjecture. It is known for some classes, including bull-free graphs, but remains open for general $H$-free graphs.