Odd-Catalan enumeration conjecture for regular triangulations of the permutohedron order polytope

Let $S_n$ denote the poset used in the paper, and let $\mathcal{O}(S_n)$ be its order polytope. Let $\operatorname{Cat}(m)$ denote the $m$th Catalan number. Odd-Catalan enumeration conjecture. The number of regular triangulations of $\mathcal{O}(S_n)$ is

$$2^{n+1}\cdot\operatorname{Cat}(2n+1).$$

The formula is motivated by an apparent relationship with odd Catalan numbers and was verified in the paper for $n=1,2,3$; it remains open in general.