The PFA superstrong-cardinal conjecture for HOD

Let $\force$ denote the Proper Forcing Axiom, let $\theta\text{-}\rm{UB}$ denote the relevant universally Baire framework, and let ${\mathbb{R}}_g$ and $\Gamma^\infty_g$ be the reals and universally Baire sets obtained from a generic $g$ as in the statement. The PFA superstrong-cardinal conjecture. Assume the Proper Forcing Axiom and let $\kappa\textgreater= \omega_2$. If $g\subseteq\rm{Coll}(\omega,\kappa)$, then

$${\rm{HOD}}^{L(\Gamma^\infty_g,{\mathbb{R}}_g)}\vDash “\text{there is a superstrong cardinal}”.$$

This is presented as a natural target for reflecting the large-cardinal complexity of the ambient universe into the HOD of a determinacy model; the supplied passage does not state whether it has been resolved.