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0^♯ from the full-forcing Σ_n axiom at ω1

Ascertain whether the Σ_n-correct forcing axiom for the class of all forcing with bound ω1, namely Σ_n–CFA_{<ω1}(all), implies the existence of 0^♯.

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Background

The author considers Σ_n–CFA for the class of all forcing, noting its potential to exceed the consequences of the corresponding maximality principle. A benchmark for large-cardinal strength in this context is whether 0♯ follows.

References

However, even the answer to the following question is unclear. Does $\Sigma_n\mhyphen CFA_{<\omega_1}(all)$ imply that $0\sharp$ exists?

— $Σ_n$-correct Forcing Axioms (2405.09674 - Goodman, 15 May 2024) in Section 3.4 (Forcing Axioms for Classes which Collapse ω1)