Dimension formula for tied Party-Hecke algebras

Let $T\mathcal{P}_n$ be the tied party monoid, and let $\mathcal{EP}_n(p,q)$ be the tied Party-Hecke algebra generated by $J_1,\ldots,J_{n-1}$, $F_1,\ldots,F_{n-1}$, and $E_1,\ldots,E_{n-1}$ with the defining relations given above. Dimension conjecture. The dimension of $\mathcal{EP}_n(p,q)$ equals the cardinality of the tied party monoid:

$$\dim(\mathcal{EP}_n(p,q))=|T\mathcal{P}_n|,$$

which is the sequence listed as OEIS A062995. This predicts that the defining algebra has a basis indexed by the elements of the tied party monoid; the parser provides no evidence that the claim has been proved or disproved.