Schur–Weyl duality for roots of unity and twisted Lawrence–Krammer–Bigelow representations

Let $\xi$ be a primitive $4m$th root of $1$, and let the $m$th twisted Lawrence–Krammer–Bigelow representation be the representation referred to below. Schur–Weyl duality conjecture. There is a Schur–Weyl duality akin to the established Schur–Weyl duality between $\mathcal{U}_\xi(\mathfrak{sl}_2)$ and the $m$th twisted Lawrence–Krammer–Bigelow representation. The paper suggests this as a direction for extending the results from the root of unity $i$ to other roots of unity; no resolution is given here.