The pathwidth–clique-number conjecture for bounded alpha-treewidth

Let $\mathcal{G}$ be a graph class. For graph parameters $\rho$ and $\omega$, say that $\mathcal{G}$ is $(\rho,\omega)$-bounded when bounded clique number in $\mathcal{G}$ implies bounded $\rho$; write $\alpha$-treewidth for the independence variant of treewidth. Pathwidth conjecture. Every $(\operatorname{pw},\omega)$-bounded graph class has bounded $\alpha$-treewidth.

This is one of two weaker statements left open after the construction of graph classes with clique-bounded treewidth and unbounded $\alpha$-treewidth; its status remains open.