The closed-conjugacy-class conjecture for Cartan subalgebras of nested ind-groups

Let $X$ be a variety, let $\mathfrak G=\bigcup_i G_i$ be a connected nested ind-subgroup of $\operatorname{Aut}(X)$, and let $T\subseteq\mathfrak G$ be a maximal torus. Let $C^0(T)$ be the connected component of the centralizer of $T$. The closed-conjugacy-class conjecture. For every $g\in C^0(T)$, the conjugacy class of $g$ is closed in $\mathfrak G$.