The Dumont permutation enumeration conjecture for pattern 4132

Let $\frak D^2_{2n}(4132)$ denote the set of restricted Dumont permutations of the second kind of length $2n$ avoiding the pattern $4132$, and let $C_n$ be the $n$th Catalan number. The conjecture concerns the cardinality of this set.

Enumeration conjecture.

$$|\frak D^2_{2n}(4132)|=C_n$$

for $n\ge 0$.

The authors note that this would agree with the corresponding enumeration for $321$-avoiding restricted Dumont permutations. The statement was presented as unproved in the paper, although the supplied context reports that it was subsequently proved by Bóna and associated work.