Symmetric-group controllability conjecture for semisimple Lie groups

Consider the bilinear system

$$\dot X=\Bigl(\sum_{i=1}^m u_iB_i\Bigr)X,\quad X(0)=I,$$

on a semisimple Lie group $G$, where $B_i$ belongs to the Lie algebra $\mathfrak{g}$ of $G$. Let $\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{k}$ be the Cartan decomposition, with $\mathfrak{h}$ the Cartan subalgebra, and let $W$ be the Weyl group of $\mathfrak{g}$. Assume that $B_i\in\mathfrak{h}$ or $B_i\in\mathfrak{k}$ for every $i=1,\dots,m$. Write $\Gamma=\{B_1,\dots,B_m\}$, set $h=\dim\mathfrak{h}$, and let $W^h$ denote the direct sum of $h$ copies of $W$.

Symmetric-group controllability conjecture. The system is controllable on $G$ if and only if there exists $\Sigma\subseteq\Gamma$ such that $\iota(\Sigma)$ is a cycle of maximal length in $W^h$.

This conjecture extends the symmetric-group characterization of controllability from systems on ${\rm SL}(3,\mathbb{C})$ to systems on general semisimple Lie groups, using the Cartan decomposition and the Weyl group. The supplied text does not state whether the conjecture has been proved or refuted.