qa.quantum algebra - Does the canonical basis of a tensor product of quantum group representations span the isotypic components of tilting modules?

It's a theorem of Lusztig that if one takes two (or more) representations of a quantum group at a non-root-of-unity choice of $q$, and look at the canonical basis of their tensor product, then each isotypic component is spanned by a subset of the canonical basis (i.e. each basis vector lies in an isotypic component) [EDIT: I mean the canonical basis of the tensor product in the sense of Lusztig's paper "Canonical bases on tensor products" not the tensor product of the canonical bases].

When one reduces this tensor product at a root of unity instead it's no longer necessarily semi-simple. But if we assume that each of the tensor factors does remain simple, it will be a tilting module, and have a canonical direct sum decomposition in "isotypic" components corresponding to tilting modules. These are quite different from the isotypic components at a generic value of $q$. Are these still spanned by canonical basis vectors?

EDIT: I would be basically equally happy if there were a split (but not canonically) filtration whose individual spaces were spanned by canonical basis vectors such that the successive quotients where tilting modules.