Conjecture on 0012-avoiding ascent sequences and Catalan bistatistics

Let ${\mathcal S}_{0012}(n)$ be the set of ascent sequences of length $n$ avoiding $0012$, let $A_{0012}(n)=|{\mathcal S}_{0012}(n)|$, let $\operatorname{fwd}$ be the length of the maximal final weakly decreasing sequence, and let $\operatorname{zeros}$ count zero entries. The 0012 Catalan-bistatistic conjecture. One has $A_{0012}(n)=C_n$, the $n$-th Catalan number; $(\operatorname{asc},\operatorname{fwd})$ on ${\mathcal S}_{0012}(n)$ has the same distribution as $(\operatorname{asc},\operatorname{RLmax})$ on 132-avoiding permutations; consequently ascents have the Narayana distribution on ${\mathcal S}_{0012}(n)$; and $(\operatorname{asc},\operatorname{fwd})$ and $(\operatorname{asc},\operatorname{zeros})$ have the same distribution on 0012-avoiding ascent sequences. The source treats these enumerative and distributional identities as conjectural and gives no resolution.