Yan and Martinez–Savage's enumeration conjecture for restricted inversion sequences

For each $n\geq1$, let $\operatorname{{\bf I}}_n$ be the set of inversion sequences

$$\operatorname{{\bf I}}_n:=\{(e_1,e_2,\ldots,e_n):0\leq e_i<i\}.$$

Let $\operatorname{{\bf I}}_n(\geq,\geq,-)$ be the subset of inversion sequences for which there do not exist $i<j<k$ such that $e_i\geq e_j\geq e_k$. Let $E_3(n)$ denote the number of partitions of $[n]$ avoiding enhanced $3$-crossings. Yan and Martinez–Savage's conjecture. The cardinality of $\operatorname{{\bf I}}_n(\geq,\geq,-)$ is $E_3(n)$. This conjecture proposes a connection between restricted inversion sequences and enhanced $3$-noncrossing partitions; the paper's stated purpose is to prove this enumeration, so the status of the conjecture should be checked against the paper's results.