Full faithfulness of the KZ functor on projective objects

Let $cKZ$ be the functor from the relevant category of modules over the affine Hecke algebra to the corresponding category of modules over the algebra $cbbK$. A projective object means a projective object in the source module category. KZ full-faithfulness conjecture. The functor $cKZ$ is fully faithful on projective objects.

This is a categorical double-centraliser-type property for the KZ functor. The candidate is not accompanied by resolution evidence, so its status remains open.