The Coxeter divisibility conjecture for nontrivial center representations

Let $\mathfrak{g}$ be a Lie algebra with Coxeter number $h$, let $\mathfrak{u}_q(\mathfrak{g})$ be the corresponding small quantum group, and let $L$ be an irreducible non-trivial representation appearing in its center $\mathbf{z}(\mathfrak{u}_q(\mathfrak{g}))$. The Coxeter divisibility conjecture. The integer $h+1$ divides the dimension of $L$:

$$h+1\mid\dim L.$$

The conjecture is based on computations outside the simply-laced case, where non-trivial representations occur; its general validity is not resolved in the supplied text.