Finiteness conjecture for Thompson groups of loop braid groups

Let $\Vloop=\Thomp{\mathit{LB}_*}$ and $\Floop=\Thomp{\mathit{PLB}_*}$ be the generalized Thompson groups containing the loop braid groups and pure loop braid groups, respectively. Finiteness conjecture. The groups $\Vloop$ and $\Floop$ are of type $\mathrm{F}_\infty$. The loop braid groups are already known to be of type $\mathrm{F}_\infty$; the conjecture therefore concerns the connectivity needed to establish the corresponding finiteness properties for these generalized Thompson groups.