Todorčević's consistency conjecture for regular hereditary Lindelöf spaces

Let $X$ be a regular hereditary Lindelöf space, abbreviated $\mathsf{HL}$. The principle $\mathsf{OGA}(X)$ denotes the open graph axiom for $X$.

Todorčević's conjecture. It is relatively consistent with $\mathsf{ZFC}$ that if $X$ is a regular $\mathsf{HL}$ space, then $\mathsf{OGA}(X)$ holds.

Under $\mathsf{PFA}$, regular spaces with countable spread that have no uncountable discrete subspace are $\mathsf{HL}$, so the source gives this as a reformulation of the broader countable-spread version. The consistency claim remains open in the source.