Optimality of Woodin's equiconsistency theorem for ordinal-definable games

An ordinal definable game of length $\omega_1$ on natural numbers with real parameters is a game in which the payoff and defining data are ordinal definable, with real parameters allowed. A game ending at $\omega_1$ in $L$ of the play is one whose play terminates at the first ordinal uncountable in the constructible universe relative to the preceding play. Optimality conjecture. If all ordinal definable games of length $\omega_1$ on natural numbers with real parameters are determined, then there is a model of ${\mathsf{ZFC}}$ with a Woodin cardinal that is a limit of Woodin cardinals. This asserts that the large-cardinal strength supplied by Woodin's equiconsistency theorem is also necessary, while the converse equiconsistency direction is known from the cited theorem; the reverse consistency-strength implication remains open.