The characterization of universally quantified visibly pushdown sets in groups

Let $G$ be a group. A set is universally quantified visibly pushdown, written $4\mathbf{VPL}^\forall4$, if it is defined by the corresponding universal visibly pushdown language construction; let $4\mathbf{REG}^\forall4$ denote the class defined analogously from regular languages. The characterization conjecture. In all groups, every $4\mathbf{VPL}^\forall4$ set is recognisable. Equivalently, the classes satisfy

$$\mathbf{VPL}^\forall=\mathbf{REG}^\forall.$$

This would identify the universally quantified visibly pushdown sets with the recognisable sets and clarify the structure of visibly pushdown constraints over arbitrary groups. The paper states that it makes significant progress towards this claim, but provides no resolution.