Speed of victory in adversarial population games

Population games involve two players, Laetitia and Terence, and an NFA with one initial and one final state. At the start, some number $N$ of tokens are placed on the initial state. At each turn, Laetitia chooses a letter $a$ and Terence moves each token along a transition labelled by $a$. The goal of Laetitia is that at some point all tokens are gathered on the final state, while Terence tries to avoid it indefinitely. It was shown in [1] that we can decide whether Laetitia wins for all number of tokens $N$. The proof uses an abstracted version of the game, called the \textbackslash{}textit{capacity game}, in which the number of tokens is not written explicitly. In the journal version [2], it was shown that if Laetitia wins then she has a strategy to win in $O(N^k)$ steps, for all $N$, with $k$ depending only on the NFA. On the other hand, we know that some games can be won in $O(log(N)^k)$ steps, or in $O(1)$ steps. We think that automata can be split in four classes: winnable in constant, polylog, polynomial number of steps, or not winnable at all. To sum up, here is what we know: It is PSPACE-complete to decide whether a game is winnable in constant time, and if not then it requires at least polylog time. It is EXPTIME-complete to decide whether a game is winnable in polynomial time, and if not then it is not winnable at all. The open problem is to complete this picture by proving the following conjecture: $\diamond$ It is decidable whether a game is winnable in polylog time, and if not then it requires at least polynomial time. Ideally, this would come with complexity bounds. [1] Bertrand, Dewaskar, Genest, Gimbert, $\textit{Controlling a population}$, CONCUR 2017 [2] Bertrand, Dewaskar, Genest, Gimbert, Godbole $\textit{Controlling a population}$, LMCS-15(3), 2019