The finiteness characterization of type $\mathfrak F\mathrm{FP}_\infty$

Let $G$ be a group. Write $\mathfrak F\mathrm{FP}_\infty$ for the finiteness property relative to the family of finite subgroups, and write $\mathrm{FP}_\infty$ for the ordinary finiteness property over the integers. A $p$-subgroup is a subgroup whose order is a power of a prime. Finiteness characterization conjecture. The group $G$ is of type $\mathfrak F\mathrm{FP}_\infty$ if and only if $G$ is of type $\mathrm{FP}_\infty$ and has finitely many conjugacy classes of $p$-subgroups. This conjecture proposes an ordinary finiteness-theoretic characterization of the family-relative property; the supplied text gives no resolution or further evidence for its status.