TARGET in asynchronous shared-memory systems

An asynchronous shared-memory system is composed of $n$ processes that interact by reading from and writing to a shared memory. This shared memory is composed of $r$ registers, each containing one symbol at a time from the finite alphabet $D$. All processes are anonymous and identical; they are described by the same finite automaton, called protocol. The transitions of the automaton are labelled by actions of the form $read_i(d)$ or $write_i(d)$ with $i \in \{1,\dots,r\}$ and $d \in D$. A step of the execution corresponds to some process taking a transition of the automaton and, doing so, reading from or writing to some register. In the initial configuration of size $n$, all $n$ processes are on state $q_0$ while all registers have initial value $d_0$. TARGET asks whether there exists $n \geq 1$ such that, from the initial configuration of size $n$, there is an execution that puts all processes on some distinguished state $q_f$. This problem is known to be NP-complete in the general case, and in PTIME when $r=1$. Questions: Is TARGET XP with respect to $r$, i.e., is it solvable in polynomial time when $r$ is fixed? Is it FPT with respect to $r$, i.e., can it be solved in time $O(f(r) p(|P|))$ with $p$ a polynomial and $P$ the protocol? We have established during Autoboz that the problem is in fct W[2]-hard with respect to r. Remain open questions are: is it FPT with respect to $r + |D|$? is it in XP with respect to $r$? in particular, is in in PTIME for $r = 2$?