The topological-rank conjecture for L^1 full groups of flows

Let $\mathcal F:\mathbb R\curvearrowright(X,\mu)$ be a measure-preserving flow, and let $n$ be the number of its ergodic components. Write $[\mathcal F]_1$ for its $\mathrm{L}^{1}$ full group and $\mathrm{rk}([\mathcal F]_1)$ for its topological rank. Topological-rank conjecture. If $\mathcal F$ has exactly $n$ ergodic components, then

$$\mathrm{rk}([\mathcal F]_1)=n+1.$$

The preceding bounds show that for a free measure-preserving flow on a standard probability space the rank lies between $n+1$ and $n+3$; the conjecture predicts that the lower bound is always attained. In particular, the rank of the full group of an ergodic flow should always be $2$.