The finiteness-length conjecture for polynomial points of simple groups

Let $K$ be a number field, let $\mathcal{O}_K$ be its ring of integers, and let $\bf H$ be a connected, noncommutative, absolutely almost simple algebraic $K$-group whose $K$-rank equals $k$.

Finiteness-length conjecture. The group

$${\bf H}(\mathcal{O}_K[t])$$

\nis not of type $F_k$ or $FP_k$.

This extends the known rank-one result that polynomial points of such groups are not finitely generated. The conjectured higher-rank finiteness obstruction is left open in the source.