Finite generation conjecture for diffeomorphism groups of closed manifolds

Let $X$ be a smooth closed manifold. Write $\mathrm{Diff}(X)$ for its diffeomorphism group, $B\mathrm{Diff}(X)$ for its classifying space, and $\pi_k$ and $H_k(-;\mathbb{Z})$ for homotopy and integral homology groups. Finite generation conjecture. If $\pi_1(X)$ is finite and $\dim X\neq 4$, then

$$\pi_k(\mathrm{Diff}(X))\text{ and }H_k(B\mathrm{Diff}(X);\mathbb{Z})$$

should be finitely generated for all $k\geq 0$. No counterexample is currently known; the conjecture concerns finiteness phenomena for diffeomorphism groups and their classifying spaces outside dimension four, under a finite fundamental-group hypothesis.