The finiteness conjecture for the exotic Picard group

Fix a prime $p$ and height $n$. Let $\mathrm{Pic}_n$ be the Picard group of the $K(n)$-local category, let $\mathrm{Pic}_n^{\operatorname{alg}}$ be its algebraic analogue, and define the exotic Picard group

$$\kappa_n:=\ker\bigl(\mathrm{Pic}_n\longrightarrow\mathrm{Pic}_n^{\operatorname{alg}}\bigr).$$

Hopkins's exotic Picard-group conjecture. The group $\kappa_n$ is a finite $p$-group. The source notes that $\kappa_n=0$ when $2(p-1)\geq n^2$, while it is nontrivial in many other cases; the stated finiteness assertion remains unresolved in general.