Conjecture on 0123-avoiding ascent sequences and bounded-height Dyck paths

Let $A_{0123}(n)$ be the number of ascent sequences of length $n$ avoiding $0123$. A Dyck path of semilength $n$ has height at most $5$ if its maximum vertical height is at most $5$. The 0123 Dyck-path conjecture. $A_{0123}(n)$ equals the number of Dyck paths of semilength $n$ and height at most $5$. The conjecture identifies the enumeration of 0123-avoiding ascent sequences with a bounded-height Dyck-path family; the source gives no resolution.