Lower bounds for determinizability of weighted automata over Q

Although quite natural, the problem of deciding if a weighted automaton can be determinized has only been solved in 2022 by Bell & Smertnig [arXiv:2209.02260]. Following this decidability result, it is shown in Benalioua, Lhote & Reynier [arXiv:2307.13505] that one can decide in 2EXPTIME if a weighted automaton over the field of rationals is equivalent to a deterministic one. It is shown in Jecker, Mazowiecki & Purser [arXiv:2310.02204] that determinizability can be decided in PSPACE, when the automata are restricted to polynomial ambiguity. Question: Can one obtain lower bounds for the decision problem of determinizability of weighted automata over $\mathbb Q$?