Burstein–Jones dominance conjecture for vincular-pattern statistics

Let $\mathfrak{D}^1_{2n}(\tau)$ be the Dumont-1 permutations of length $2n$ avoiding $\tau$. For a permutation $\pi$, let $(2\text{-}31)\pi$ and $(13\text{-}2)\pi$ be the numbers of occurrences of the indicated vincular patterns, where the dash indicates that the corresponding adjacent pattern letters need not be adjacent in $\pi$. Define

$$a_{n,k}=|\{\pi\in\mathfrak{D}^1_{2n}(2143)\mid (2\text{-}31)\pi=k\}|,$$ $$b_{n,k}=|\{\pi\in\mathfrak{D}^1_{2n}(3421)\mid (13\text{-}2)\pi=k\}|.$$

Burstein–Jones's dominance conjecture. For all $n\geq0$,

$$a_{n,k}=b_{n,k}=1\quad\text{for }k=\binom{n}{2},\qquad a_{n,k}=b_{n,k}=0\quad\text{for }k>\binom{n}{2},$$

and

$$\sum_{k=0}^{m}a_{n,k}\geq\sum_{k=0}^{m}b_{n,k}\quad\text{for }0\leq m\leq\binom{n}{2},$$

with equality when $m=\binom{n}{2}$. This refines the Wilf-equivalence conjecture by comparing distributions of vincular-pattern statistics; the source presents it as open and reports no proof beyond partial inroads.