Maximal-chain conjecture for the subgroups of Thompson's group F

Let $\mathfrak{F}$ be the partially ordered family of relevant subgroups under embeddability, let $\varepsilon_0$ be the ordinal used to index the chain, and extend the chain by setting $G_{\varepsilon_0}=F$. Maximal-chain conjecture.

$$\{G_\xi \mid \xi \leq \varepsilon_0\}$$

is a maximal chain in $\mathfrak{F}$. The conjecture proposes a complete maximal chain of subgroup-complexity representatives indexed through $\varepsilon_0$; its status is not resolved in the supplied source.