Non-semistability and P$3$ R-groups

One-ended groups which are semistable are known to be P$3$ R if and only if the fundamental pro-group is pro-isomorphic to a tower of finitely generated free groups of increasing rank, where the bonding maps are projections, see [CLQ06]. It is an open question what kind of towers may occur as fundamental pro-groups of (non-semistable) P$3$ R-groups. Notice that the possible proper $3$ -realizations for these groups are open Whitehead manifolds, once the boundary is filled up by spheres and semiplanes if necessary. It is also unknown whether or not the class of P$3$ R-groups is contained in the class of semistable groups.