Membership in Reversed Partially-Ordered Automata

Let $A$ be an automaton and write, for any word $w$, $f_w$ for the function that maps any state $q$ to the state reached by reaching $w$ from $q$. The membership problem MEMB($V$) for a class $V$ of deterministic automata is the following: Given: An automaton $A \in V$ and a function $f\colon Q \to Q$ from states to states Question: Is there a word $w$ such that $f_w = f$? In the late 1980s, Beaudry, together with McKenzie and Thérien, studied the problem for most natural classes $V$ (e.g., groups and aperiodic automata). Quoting Beaudry, McKenzie, Thérien, 1992: "For each $V$ aperiodic, the computational complexity of MEMB($V$) turns out to look familiar. Only five possibilities occur as $V$ ranges over the whole aperiodic sublattice: With one family of NP-hard exceptions whose exact status is still unresolved, any such MEMB($V$) is either PSPACE-complete, NP-complete, P-complete or in AC$^0$." The open case is with $V$ defined in any of the following equivalent ways: The set of minimal automata such that the reverse language is recognized by a partially-ordered automaton (i.e., the transition function induces a partial order). The set of minimal automata in which for every word $w$, if $f_w = f_{ww}$ then $f_w = f_{aw}$ for any letter $a$ of $w$. The set of minimal automata of $\mathcal{L}$-trivial languages. Question: Is the membership problem for that class in NP or PSPACE-hard? (These are the two possibilities.)