Affine Schur–Weyl reciprocity over a non-root-of-unity specialization

Let $n$ and $r$ be as in the affine quantum Schur–Weyl construction, let ${\boldsymbol{\mathcal H}_{\!\vartriangle\!}(r)}$ and ${\boldsymbol{\mathcal S}_{\!\vartriangle\!}({n^{\!\circ\!}},r)}$ be the affine Hecke and quantum Schur algebras, and let $\boldsymbol\Omega^{\otimes r}$ be the tensor space. Write $(-)_\mathbb C$ for base change obtained by specializing $v$ to a non-root of unity $z\in\mathbb C$. Affine Schur–Weyl reciprocity conjecture. The algebra homomorphisms

$$\xi_r^\vee:{\boldsymbol{\mathcal H}_{\!\vartriangle\!}(r)}\longrightarrow\operatorname{End}_{{\boldsymbol{\mathcal S}}_{\!\vartriangle\!}({n^{\!\circ\!}},r)}(\boldsymbol\Omega^{\otimes r}),\qquad\xi_{r,\mathbb C}^\vee:{{\mathcal H}_{\!\vartriangle\!}(r)}_\mathbb C\longrightarrow\operatorname{End}_{{\mathcal S}_{\!\vartriangle\!}({n^{\!\circ\!}},r)_\mathbb C}(\Omega_\mathbb C^{\otimes r})$$

are surjective. The conjecture would establish affine Schur–Weyl reciprocity over $\mathbb Q(v)$ and over $\mathbb C$ for a non-root-of-unity specialization; the source says surjectivity over $\mathbb Q(v)$ is unknown.