Noncommuting one-parameter subgroups conjecture for continuous interval exchange actions

Let $\mathcal{E}$ be the topological group of continuous interval exchange actions, and let $F=\{f_t\}$ and $G=\{g_s\}$ be one-parameter subgroups of $\mathcal{E}$. Noncommuting one-parameter subgroups conjecture. If $F$ and $G$ do not commute, then the group $\langle F,G\rangle$ has elements that are not contained in any one-parameter subgroup of $\mathcal{E}$. The commuting case is described by simultaneous conjugacy to subgroups of a common standard torus action; the conjecture asserts that noncommuting one-parameter subgroups exhibit fundamentally different behavior. Its resolution is not indicated in the source.