The group-pair analogue of the PD3 splitting theorem

Let $(G,\mathbf S)$ be a $it{PD}^3$ group pair in the setting of the preceding splitting theorem.

Group-pair splitting conjecture. There is an analogue for group pairs of the preceding theorem on $it{PD}^3$ groups.

The conjecture is proposed to remove the hypothesis Max--c, meaning the maximal condition on centraliser subgroups, from the corresponding decomposition theorem. The source says that the question might be resolvable by the methods used for the theorem, but gives no resolution.