Smooth Schubert enumeration conjecture for signed permutations

Let $S_n^B$ be the group of signed permutations of size $n$, and let $\operatorname{GAV}_n(\{3412,4231\})$ denote the signed permutations in $S_n^B$ that globally avoid the patterns $3412$ and $4231$. Smooth Schubert varieties in type A are indexed by permutations avoiding $3412$ and $4231$.

Smooth Schubert enumeration conjecture. The number of signed permutations in $S_n^B$ that simultaneously index smooth Schubert varieties in types B and C is equal to the number of permutations in $S_{n+1}$ that index a smooth type-A Schubert variety; equivalently,

$$\left|\operatorname{GAV}_n(\{3412,4231\})\right| =\left|\{w\in S_{n+1}:w\text{ avoids }\{3412,4231\}\}\right|.$$

The conjecture is motivated by computations for small $n$ and predicts an equality between a type-B/C signed-permutation enumeration and the classical type-A smooth-Schubert enumeration. Its status is unresolved in the supplied source.