Undecidability of regular winning-position languages for strongly terminating games

A strongly terminating game is a game $\mathcal{G}$ in which every play terminates. The winning positions of $\mathcal{G}$ are the positions from which the winning player can force a win; these positions form a language over the game's position alphabet.

The conjecture. Given a strongly terminating game $\mathcal{G}$, it is undecidable whether the winning positions of $\mathcal{G}$ form a regular language.

The paper presents this as an open problem in the context of taking-and-merging games. The preceding results establish undecidability for related restricted questions, but the supplied text does not give a proof of this stronger regularity problem.