Determine the exact balanced-product formula for two-color critical multiplicity

Prove or disprove the conjectured exact formula \(m_2(s)=\prod_{j=0}^{s-3}\left\lceil\frac{\left\lfloor(R_2(s)-2)/2\right\rfloor-\left\lceil(s-1)/2\right\rceil-j}{s-2}\right\rceil\) for every integer s in the two-color Ramsey problem.

Background

The second conjecture strengthens the preceding set-valued conjecture by assuming that the two monochromatic subgraphs generated when the final edge is colored have sizes as equal as possible. The smaller subgraph is then assigned the proposed minimum size, and the counting inequality is assumed to be attained with equality.

The authors explicitly characterize this conjecture as speculative and potentially false, although they state that it agrees with the few cases currently known. If true, it would give a precise formula rather than merely an upper bound for the critical multiplicity m_2(s).

References

The third item is arguably less reasonable but, all the same, it yields the following (probably, reckless) conjecture that we expect might be wrong, but is correct in the very few cases that are known. It follows by taking the value in the third item and applying it to \Cref{bound theorem}, obtaining equality by the second item, and obtaining a third equality by the first item.

m_2(s) = \prod_{j=0}{s-3}\left\lceil\frac{\left\lfloor\frac{R_2(s)-2}{2}\right\rfloor - \left\lceil \frac{s-1}{2} \right\rceil -j}{s-2}\right\rceil

— Bounds on the Critical Multiplicity of Ramsey Numbers with Many Colors  (2501.18869 - Christopherson et al., 31 Jan 2025) in Section 3, “Concluding Remarks,” Conjecture 2