Equidistribution of statistics on Stoimenow matchings, Fishburn posets and Dyck paths

Let $\mathcal{M}_n(P_1)$ be the set of Stoimenow matchings avoiding $P_1$, let $\mathcal{P}_n{\bf (3+1)}$ be the corresponding class of $(3+1)$-avoiding Fishburn posets, and let $\mathcal{D}_n$ be the set of Dyck paths. Write $\operatorname{\text{\sf{nr}}}$ for the statistic on $\mathcal{M}_n(P_1)$ and $\operatorname{\text{\sf{h}}}$ for the statistics on $\mathcal{P}_n{\bf (3+1)}$ and $\mathcal{D}_n$. Equidistribution conjecture. For every $n$, these statistics have the same distribution:

$$\sum_{M \in \mathcal{M}_n(P_1)} x^{\operatorname{\text{\sf{nr}}}(M)} = \sum_{P \in \mathcal{P}_n{\bf (3+1)}} x^{\operatorname{\text{\sf{h}}}(P)} = \sum_{\mu \in \mathcal{D}_n} x^{\operatorname{\text{\sf{h}}}(\mu)}.$$

This conjecture extends the equidistribution result proved in Theorem~; it predicts a common generating polynomial for the three families, while the general equality remains open in the source.