Let $\mathfrak g$ be a simple Lie algebra.

By taking the specialization at $q^\ell=1$ of a certain integral version¹ of the quantum group $U_q(\mathfrak g)$,
and by considering a certain quotient category² of the category of tilting modules³ over that Hopf algebra, one obtains a fusion category.
Moreover, by using the universal $R$-matrix and the so-called charmed element, one can endow that fusion category with the structure of a modular tensor category.

Can this approach be used to further equip this category with the structure
of a **unitary** modular tensor category?

In other words, is it possible to equip the objects of this category with some kind of extra structure (something like an inner product) such that the adjoint $f^*:W\to V$ of a morphism $f:V\to W$ makes sense, and such that the braiding and twist are unitary?

¹: That integral version is usually denoted $U_q^{res}(\mathfrak g)$. It is generated by the usual elements $K_i$, $E_i$, $F_i$, along with the divided powers $E_i^{[r]}:=\frac{E_i^r}{[r]!}$ and $F_i^{[r]}:=\frac{F_i^r}{[r]!}$.

²: This quotient category has the same objects as the original category. The hom-spaces are modded out by the subspace of negligible morphisms, where a morphisms $f:V\to W$ is called negligible if $tr_q(fg)=0$ for any $g:W\to V$.

³: A module is tilting if it admits a filtration whose associated graded pieces are Weyl modules, and also admits a filtration whose associated graded pieces are duals of Weyl modules. Here, the Weyl modules, are the ones that "look like" irreps of $\mathfrak g$.