The finiteness-property conjecture for kernels in Lodha--Moore groups

Let $m\in\mathbb{N}\cup\{\infty\}$, let $\psi$ be the character considered in the surrounding discussion, and let the ambient group be one of the Lodha--Moore groups $G$, $G_y$, ${}_yG$, or ${}_yG_y$. Finiteness-property conjecture. The kernel of $\psi$ in some Lodha--Moore group is of type $\operatorname{F}_m$. This conjecture would yield a full computation of the BNSR-invariants for all four Lodha--Moore groups.