Burstein–Jones Wilf-equivalence conjecture for Dumont-1 permutations

Let $\mathfrak{D}^1_{2n}$ denote the set of Dumont-1 permutations of length $2n$, and let $\mathfrak{D}^1_{2n}(\tau)$ denote those avoiding the pattern $\tau$. Burstein–Jones's Wilf-equivalence conjecture. For all $n\geq 0$,

$$|\mathfrak{D}^1_{2n}(2143)|=|\mathfrak{D}^1_{2n}(3421)|.$$

The conjecture asserts that the two avoidance sequences are equal for every $n$. They agree computationally through $n=10$, but the source says that the conjecture is difficult to prove and only partial progress was known.