Poly-context-free characterization of lattices in products of trees

Let $\Gamma$ be a lattice in a product of trees. It is reducible if it is virtually a direct product of free groups. A group is poly-context-free if its word problem is an intersection of finitely many context-free languages.

Poly-context-free lattice conjecture. A lattice in the product of trees is poly-context-free if and only if it is reducible.

The reducible case is poly-context-free because finite direct products of free groups are poly-context-free and the class is stable under the relevant virtual operations. The converse is suggested by the paper's result for the quaternionic lattices, but is stated there as a possible generalization and remains open.