Finiteness equivalence for Thompson-type groups of a Cantor-algebra

Let $U_r(\Sigma)$ be a Cantor-algebra, with associated groups $G_l(\Sigma)$ for $1\leq l\leq d$. The notation $\operatorname{FP}_\infty$ and $\operatorname{F}_\infty$ denotes the corresponding ordinary finiteness properties, while quasi-$\underline{\operatorname{FP}}_\infty$ and quasi-$\underline{\operatorname{F}}_\infty$ denote their Bredon finiteness analogues.

Finiteness equivalence conjecture.

$$G_r(\Sigma)\text{ is quasi-}\underline{\operatorname{FP}}_\infty\Leftrightarrow G_l(\Sigma)\text{ is of ordinary type }\operatorname{FP}_\infty\text{ for any }1\leq l\leq d,$$

and

$$G_r(\Sigma)\text{ is quasi-}\underline{\operatorname{F}}_\infty\Leftrightarrow G_l(\Sigma)\text{ is of ordinary type }\operatorname{F}_\infty\text{ for any }1\leq l\leq d.$$

The surrounding text presents closely related equivalences as theorems for the groups $T_r(\Sigma)$, but the supplied candidate is marked with unknown status and its notation differs from that theorem statement; its precise conjectural status and intended relation to the proved results should be checked against the source.