Strong deformation retract conjecture for twisted Hopf link character varieties

Let $H_n$ be a twisted Hopf link, and let $X(H_n,G)$ denote its $G$-character variety. For an integer $r\geq 1$, consider the character varieties for $G=\operatorname{SU}(r)$ and $G=\operatorname{SL}(r,\mathbb{C})$. Strong deformation retract conjecture. For any $r\geq 1$, the $\operatorname{SU}(r)$-character variety of a twisted Hopf link is a strong deformation retract of the corresponding $\operatorname{SL}(r,\mathbb{C})$-character variety. The conjecture proposes a higher-rank extension of the established rank-two deformation-retract result. The paper gives detailed semisimple stratifications in the $\operatorname{U}(2)$ and $\operatorname{SU}(3)$ cases, but the complicated intersection patterns prevent the further cohomological calculations needed to determine the general homotopy type.