The equality of the sharp and recursive Potts constants

Let $q$ be the number of Potts symbols, let $H$ and $R g$ be the functions defined from the Potts recursion, and define

$$\hat c(\beta,q):=\sup_{p\in P,\,p_2=\dots=p_q}\frac{H(p_1,\dots,p_q)}{R g(p_1,\dots,p_q)}.$$

Here $\bar c(\beta,q)$ is the constant appearing in the paper's linear recursion bound.

Equality conjecture. We believe that

$$\hat c(\beta,q)=\bar c(\beta,q).$$

This conjecture concerns whether the variational constant obtained by restricting to vectors with equal coordinates $p_2=\dots=p_q$ coincides with the constant used in the reconstruction estimate. The source gives no resolution of this claim.