Hu's surjectivity conjecture for the symplectic Schur–Weyl map

Assume that $K$ is an algebraically closed field, let $V$ be the symplectic vector space underlying the action of $KSp(V)$, and let $\mathfrak{B}_{n,K}$ be the Brauer algebra with ideals $\mathfrak{B}_{n,K}^{(f)}$. For an integer $f$ with $0\leq f\leq [n/2]$, consider the natural algebra homomorphism

$$\psi_{f,K}:KSp(V)\longrightarrow \operatorname{End}_{\mathfrak{B}_{n,K}/\mathfrak{B}_{n,K}^{(f)}}\bigl(V^{\otimes n}/V^{\otimes n}\mathfrak{B}_{n,K}^{(f)}\bigr).$$

Hu's surjectivity conjecture. The map $\psi_{f,K}$ is surjective. This conjecture asks for the symplectic double-centralizer property on the quotient by the $f$-th Brauer ideal; it is presented as an original starting point of the paper's work and is not stated there as proved.