Conjecture on Wilf equivalence of 0021 and 1012

For a pattern $p$, let $A_p(n)$ be the number of ascent sequences of length $n$ avoiding $p$. Two patterns are Wilf equivalent if their avoidance sequences agree for every $n$. The 0021–1012 Wilf-equivalence conjecture. The patterns $0021$ and $1012$ are Wilf equivalent, and $A_{0021}(n)=A_{1012}(n)$ is given by the binomial transform of the Catalan numbers. The source gives no resolution of this proposed equivalence and enumeration.