Projectivity conjecture for finitely generated pro-$\mathcal{C}$ subgroups

Let $G$ be a torsion-free hyperbolic virtually compact special group, let $\widehat{G}$ be its profinite completion, and let $H$ be a finitely generated pro-$\mathcal{C}$ subgroup of $\widehat{G}$, where $\mathcal{C}$ does not contain all finite groups.

Projectivity conjecture. $H$ is projective.

This proposes projectivity for finitely generated pro-$\mathcal{C}$ subgroups arising in profinite completions of torsion-free hyperbolic virtually compact special groups. The supplied text does not state whether the conjecture is known or remains open.