The genus-2 handlebody representation conjecture for the Clifford orbit of $W_2$

Let $W_2$ denote the two-anyon-pair state, and let $\Sigma_2$ be a genus-$2$ handlebody. The Clifford group orbit of $W_2$ is represented by removing two solid tori from the interior of $\Sigma_2$. Handlebody representation conjecture. A Dehn twist is implemented on each genus-$1$ component of $\Sigma_2$. This conjecture proposes a topological realization of Clifford evolution in $SU(2)_1$ Chern–Simons theory, relating the Clifford orbit of the state to Dehn-twist operations on the corresponding genus-$2$ handlebody. The supplied text does not state whether the conjecture has been proved or disproved.