A counterexample to the pure semisimplicity conjecture

A ring is right pure semisimple if every right module is a direct sum of finitely generated modules, and the pure semisimplicity conjecture asserts that every such ring is of finite representation type. In this article, a division ring extension $H\subseteq T$ is constructed such that the bimodule ${}_HT_T$ has right dimension sequence $(1,2,2,2,\dots)$ and $T$ is infinite-dimensional as a left $H$-space. The division ring $T$ is the union of the fraction division rings of the nearest-neighbour quantum Laurent algebras with parameter $1/2$, and $H$ is the image of $T$ under a compatible family of conjugations. The extension is controlled by right bases of every length and by two-sided flags of adjacent division subrings, which identify all iterated right duals up to explicit twists. It follows that the triangular ring $\left(\begin{smallmatrix}H&T\\0&T\end{smallmatrix}\right)$ is right pure semisimple but neither left pure semisimple nor of finite representation type, so that the pure semisimplicity conjecture fails.