Exact product formula for two-color critical multiplicity

Prove that, for every integer s, the two-color critical multiplicity satisfies the exact formula $$m_2(s)=\prod_{j=0}^{s-3}\left\lceil\frac{\left\lfloor\frac{R_2(s)-2}{2}\right\rfloor-\left\lceil\frac{s-1}{2}\right\rceil-j}{s-2}\right\rceil.$$

Background

The paper derives upper bounds for m_2(s) by applying a counting inequality to the monochromatic K_s copies containing a final edge. A stronger formula is proposed by assuming that the two associated monochromatic subgraphs have nearly equal sizes, that their noncommon vertices partition the complete graph, and that both attain the counting bound.

The resulting product formula is explicitly described as potentially incorrect, although it agrees with the few cases known to the authors. Establishing it would give a precise value of m_2(s) rather than merely an upper bound.