The center-trace equality conjecture for factorizable ribbon Hopf algebras

Let $A$ be a finite-dimensional, unimodular, ribbon, $\Lambda$-factorizable algebra. Let $\mathcal T$ be the subset of elements of the symmetric quotient of the center that define invariants of 4-thickenings under 2-deformations, and let $\mathcal T_Z$ be the corresponding center-restricted subset.

Center-trace equality conjecture.

$$\mathcal T_Z=\mathcal T.$$

For the quantum $sl(2)$ case, the paper finds four elements of $\mathcal T_Z$ up to multiplication by a scalar, with three lying in the link-invariant subset. The conjecture asserts the equality for every algebra satisfying the stated hypotheses.