Jordan decomposition conjecture for factorizable ribbon quantum groups

Let $H$ be a factorizable ribbon quantum group, with ribbon element having multiplicative Jordan decomposition. The decomposition gives $SL(2,\mathbb{Z})$-representations $\bar\pi$ and $\pi^*$ satisfying the properties of Theorem 2.2: they commute pointwise, the original representation factors as $\pi(\gamma)=\bar\pi(\gamma)\pi^*(\gamma)$, and $\bar\pi$ restricts to the image of the Grothendieck ring in the center. Jordan decomposition conjecture. For any factorizable ribbon quantum group, the multiplicative Jordan decomposition of its ribbon element gives rise to these $SL(2,\mathbb{Z})$-representations with all three stated properties. This extends the construction established for the restricted quantum group associated with $s\ell(2)$, and remains conjectural in general.