Conjecture on the kernel quotient map for specialized BMW algebras

For $n\geq 3$, let $\mathcal{B}_n^{\pm}$ be the specialized algebra, let $\mathcal{I}_n^{\pm}$ be its defining ideal, and let $\pi_n^{\pm}:\mathcal{I}_n^{\pm}\to\mathcal{J}_n^{\pm}$ be the leftmost vertical map in the commutative diagram relating $\mathcal{B}_n^{\pm}$ to the quotient of $\mathbbm{Q}\mathfrak{S}_n$ by $\mathcal{J}_n^{\pm}$. Kernel quotient-map conjecture. The map

$$\pi_n^{\pm}$$

is an isomorphism for all $n\geq 3$. The computations establish this for $n=3$ and $n=4$, while the general case is left open.