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Separation in the ccc case: MP_ccc+RRP without CMA

Determine whether it is consistent that the Σ_2 maximality principle for ccc forcing Σ_2–MP_{ccc}(H_𝔠) and the residual reflection principles Σ_2–RRP(𝔠,λ,ccc) hold for all λ≥𝔠 while the Σ_2‑correct Martin’s Axiom Σ_2–CMA fails.

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Background

This is a concrete instance of the broader separation problem in the important ccc context. A positive answer would show that Σ2–CBMA (Σ_2–CMA) is strictly stronger than MP{ccc}+RRP at H_𝔠, clarifying the landscape for ccc forcing.

References

Further open questions arise from the fact that, although the proof of Theorem \ref{thm:cbfafactor} does not appear to generalize beyond the provably self-preserving classes, it is unclear how to establish the separations that would definitively show that the theorem does not generalize: Is it consistent that $\Sigma_2\mhyphen MP_{ccc}(H_\mathfrak{c})$ and $\Sigma_2\mhyphen RRP(\mathfrak{c},\lambda, ccc)$ hold for all $\lambda\geq \mathfrak{c}$, but $\Sigma_2\mhyphen CMA$ fails?

— $Σ_n$-correct Forcing Axioms (2405.09674 - Goodman, 15 May 2024) in Section 7 (Residual Reflection Principles), end of section