Conjecture on 210-avoiding ascent sequences and 3-noncrossing partitions

Let $A_{210}(n)$ be the number of ascent sequences of length $n$ avoiding $210$. A set partition of $\{1,2,\ldots,n\}$ is 3-noncrossing if it has no 3-crossing. The 210 counting conjecture. $A_{210}(n)$ equals the number of 3-noncrossing set partitions of $\{1,2,\ldots,n\}$. This is the formal counting version of the earlier prose conjecture relating 210-avoiding ascent sequences to partitions avoiding 3-crossings; the source gives no resolution.