Ostrik's semisimplicity conjecture for quantum doubles

Let $\mathfrak{g}$ be a semisimple Lie algebra, and let $U_q(\mathfrak{g})$ be its quantized enveloping algebra with quantum double $D(U_q(\mathfrak{g}))$. A finite-dimensional representation is a finite-dimensional left module over this double. Ostrik's semisimplicity conjecture. Every finite-dimensional representation of $D(U_q(\mathfrak{g}))$ is semisimple. This was proposed for generic $q$ in conversations with Victor Ostrik and is described as an open and difficult problem, even for $\mathfrak{g}=\mathfrak{sl}_2$.