Wonderful compactification simple connectedness conjecture for exponent canceling groups

Let $\Gamma$ be an exponent canceling group, meaning that its abelianization is free abelian, and let $G$ be a connected adjoint semisimple algebraic group over $\mathbb{C}$. Write $\mathfrak{X}^0_\Gamma(G)$ for the component of the character variety containing the trivial representation, and let its normalized wonderful compactification be the corresponding normalized wonderful compactification.

Wonderful compactification conjecture. The normalized wonderful compactification of $\mathfrak{X}^0_\Gamma(G)$ should be a simply connected compactification of $\mathfrak{X}^0_\Gamma(G)$.

The conjecture extends the simple connectedness expectation for character varieties to their normalized wonderful compactifications. The paper proves this property in some cases, including free abelian groups with suitable semisimple adjoint groups and surface groups with $G=\mathrm{PGL}_n$, but the stated general claim remains open.