Cerbai–Claesson–Ferrari bijection conjecture for 201-avoiding ascent sequences

Let $A_n(201)$ be the set of ascent sequences of length $n$ avoiding the pattern $201$, and let $S_n$ be the set of permutations of length $n$. A permutation is $312$-machine sortable if it can be sorted using two series-connected stacks whose first stack avoids $312$ when read from top to bottom and whose second stack is increasing.

Cerbai–Claesson–Ferrari conjecture. For every $n\geq 0$, there is a bijection between $A_n(201)$ and the permutations in $S_n$ that are $312$-machine sortable.

The conjecture is supported by data reported by Cerbai, Claesson, and Ferrari, but the source gives no proof.