Optimality of Neeman's continuously coded length determinacy theorem

Let $G_{\operatorname{cont}}(\nu,A)$ be the game of continuously coded length, where $\nu$ is a partial function from reals to natural numbers and $A$ is a payoff set of sequences of reals. The notation $\boldsymbol\Sigma^0_2$ denotes the relevant Borel pointclass, and ${<}\omega^2-\boldsymbol\Pi^1_1$ in the codes means that the set of codes for elements of $A$ belongs to that pointclass. Optimality conjecture. If the game $G_{\operatorname{cont}}(\nu,A)$ is determined for every $\nu$ in the class $\boldsymbol\Sigma^0_2$ and every $A$ that is ${<}\omega^2-\boldsymbol\Pi^1_1$ in the codes, then there is a model of ${\mathsf{ZFC}}$ with a Woodin cardinal $\delta$ and a cardinal $\kappa<\delta$ that is $(\delta+1)$-strong. This would show that the large-cardinal hypothesis in Neeman's determinacy theorem is optimal, complementing the known forward implication; the exact consistency strength of the determinacy assumption remains conjectural.