This is a follow-up to the last part of an old MSE answer of mine. Briefly, an analogue at $\omega_1$ of Steel's equivalence between clopen and open determinacy can be proved assuming $\mathsf{CH}$, and I'm curious whether that's necessary.
Let $C_{\omega_1}$ be the set of subtrees of ${\omega_1}^{<{\omega_1}}$ thought of as open games on ${\omega_1}$ of length ${\omega_1}$: there are two players $1$ and $2$ who alternately play elements of $\omega_1$ for $\omega_1$-many rounds with player $1$ moving at limit rounds, the first player to "fall off" the tree loses, and if the play never leaves the tree then player $2$ wins. Let $C_{\omega_1}^{1?}$ be the subset of $C_{\omega_1}$ consisting of games in which player $2$ does not have a winning strategy, let $D_{\omega_1}$ be the subset of $C_{\omega_1}$ consisting of trees with no length-${\omega_1}$ path (the "${\omega_1}$-clopen" games), and let $D_{\omega_1}^{1?}=C_{\omega_1}^{1?}\cap D_{\omega_1}$.
It's not hard to show (see below) that assuming CH we can "continuously" reduce open games on $\omega_1$ to clopen games on $\omega_1$, in the following sense. Let $X$ be the set of all functions $\omega_1^{<\omega_1}\rightarrow\omega_1$; both games and strategies can be represented by elements of $X$. Let $\tau$ be the topology on $X$ generated by sets of the form $\{f: f\supseteq b\}$ where $b$ is a partial function $\omega_1^{<\omega_1}\rightarrow\omega_1$ with countable domain. Then there are $\tau$-continuous-on-their-domains functions $F,G$ such that
$F: C_{\omega_1}^{1?}\rightarrow D_{\omega_1}^{1?}$, and
whenever $T\in C_{\omega_1}^{1?}$ and $\Sigma$ is a winning strategy for player $1$ in $F(T)$, $G(\Sigma)$ is defined and is a winning strategy for player $1$ in $T$. (This is a bit abusive, but coding maps $\omega_1^{<\omega_1}\rightarrow\omega_1$ by subsets of $\omega_1^{<\omega_1}$ is straightforward.)
For brevity, I'll write "$(+)$" for the statement that such a pair $F,G$ exists. (Essentially, this is a kind of "$\omega_1$-boldface Weirauch reduction" between the "Win for $1$" problems for open and clopen games on $\omega_1$ of length $\omega_1$. I don't know it's exact relationship to the framework proposed in Osso's recent M.Sc. thesis, however.)
I'm curious about the role of cardinal arithmetic here.
Question: Is $\mathsf{ZFC}+\neg(+)$ consistent?
I'm especially interested in whether $\mathsf{ZFC+PFA}\vdash\neg(+)$.
(An even more ambitious question is, "Is 'Every game in $D_{\omega_1}(2)$ is determined but not every game in $C_{\omega_1}(2)$ is determined' consistent with $\mathsf{ZF}$ relative to large cardinals?," where the "$(2)$"-suffix means that the moves are from $\{0,1\}$ as opposed to $\omega_1$, but I suspect that's intractable. Also, there's nothing very special about $\omega_1$ here, the argument below works for all infinite cardinals $\kappa=\kappa^{<\kappa}$, but looking past $\omega_1$ doesn't seem to make things easier.)
Proof of $(+)$ assuming CH
Fix some length-$\omega_1$ enumeration $(\sigma_\eta)_{\eta<\omega_1}$ of $\omega_1^{<\omega_1}$; I'll think of the $\sigma_\eta$s as possible plays by player $1$ in a given game. For $T\in C_{\omega_1}^{1?}$, let $S_T$ be the tree of "not-yet-losing" strategies for player $2$ in $T$; basically, a node of height $\lambda$ in $S_T$ is a map $\{\sigma_\eta:\eta<\lambda\}\rightarrow\omega_1$ which does not yet lose to any $\sigma_\eta$ with $\eta<\lambda$. Since $T\in C_{\omega_1}^{1?}$, the tree $S_T$ has no length-$\omega_1$ branch. Let $P_T$ be the poset of chains in $S_T$, ordered by end extension. $P_T$ as well has no increasing $\omega_1$-chains, and moreover by essentially the same argument as Hartogs' theorem (see also Kurepa, Ensembles ordonnes et leurs sous-ensembles bien ordonnes, Theorem 1) there is no embedding of $P_T$ into $S_T$.
Now given $T\in C_{\omega_1}^{1?}$, let $\widehat{T}$ be the game which is played like $T$ but player $1$ additionally builds an increasing sequence through $P_T$. We then have:
The construction $T\mapsto\widehat{T}$ is continuous with respect to $\tau$.
$\widehat{T}\in D_{\omega_1}$, since a path through $\widehat{T}$ would yield a winning strategy for $2$ in $T$.
Additionally, since $P_T$ doesn't embed into $S_T$, a winning strategy for $2$ in $\widehat{T}$ would yield a winning strategy for $2$ in $T$ itself, so $\widehat{T}\in D_{\omega_1}^{1?}$. (This isn't totally trivial; I can add details if folks want.)
Finally, a winning strategy $\Sigma$ for $1$ in $\widehat{T}$ yields a winning strategy $\widehat{\Sigma}$ for $1$ in $T$ just by "forgetting" the additional data, and the map $\Sigma\mapsto\widehat{\Sigma}$ is clearly continuous with respect to $\tau$.
But this uses CH in a pretty essential (and repeated) way, so I don't see how it helps in $\mathsf{ZFC}$ alone.
