Target for pushdown RBN

Reconfigurable Broadcast Networks (RBN) are a model for large groups of identical agents communicating via unreliable broadcast. A pushdown RBN (PRBN) is simply one where each agent is modeled by a pushdown transition system. A PRBN on a message alphabet $\mathcal{M}$ is described by a pushdown automaton $\mathcal{A}$ over the alphabet $\{br(m), rec(m) | m \in \mathcal{M} \}$. Configurations of $\mathcal{A}$, i.e., pairs $(q, \sigma)$ of state and stack content, are called local configurations. A configuration of the PRBN is a function from a finite set of agents $\mathbb{A}$ to local configurations. A run starts with an arbitrarily large set of agents, all in the initial state with an empty stack . A step consists of one agent taking a transition with a broadcast $br(m)$, and each other agent either not moving or taking a transition $rec(m)$ (in other words, one agent broadcasts and other agents non-deterministically receive the broadcast or not). Classic problems on such models ask whether a given set of configurations is reachable. A particular case of interest is whether we can reach a configuration where all agents are in a given state $q_f$. This problem is called Target. The open problem is to find tight complexity bounds on the Target problem for PRBN. The problem on finite-state RBN is known to be solvable in polynomial time. An easy reduction from Horn satisfiability shows PTIME- completeness. On the other hand, one can show that the problem for PRBN is in NP: intuitively, to witness Target, it suffices to one can guess the set of messages $\mathcal{M}' \subseteq \mathcal{M}$ of messages used, and two orders on this set, an order of appearance $\leq_a$ and one of disappearance $\leq_d$. Those orders are the one in which messages are first broadcast in the run, and the one in which messages are last broadcast. Then one checks, for each $m \in \mathcal{M}'$, that there are local runs that broadcast $m$ and only receive messages lower for $<_a$ beforehand (resp. greater for $<_d$ afterwards). If they exist, one can construct a run witnessing Target by making many agents follow each of those runs. If a run witnessing Target exists, they can be obtained by taking the local runs of agents broadcasting first and last each message. Is the Target problem for PRBN NP-hard? in PTIME?