Hesselholt–Rains conjecture on torsion in preprojective algebras

Let $n\geq 2$ and let

$$\Pi:= \mathbb{Z}\langle x_1,\ldots,x_n,y_1,\ldots,y_n\rangle\big/\left(\sum_{i=1}^n[x_i,y_i]\right),$$

with $\Pi_{\mathrm{cyc}}:=\Pi/[\Pi,\Pi]$. Set $r:=\sum_{i=1}^n[x_i,y_i]$, and for each prime $p\geq 2$ and $\ell\geq 1$ let $r^{(p^\ell)}$ be the class defined by dividing the image of $r^{p^\ell}$ by $p$ in the cyclic quotient of the free algebra and then mapping to $\Pi_{\mathrm{cyc}}$.

Hesselholt–Rains conjecture. For every prime $p\geq 2$, the classes $r^{(p^\ell)}$ are nonzero and generate the $p$-torsion of $\Pi_{\mathrm{cyc}}$. Moreover, $\Pi_{\mathrm{cyc}}$ has no $p^2$-torsion.

The conjecture concerns the torsion in zeroth Hochschild homology of the simplest preprojective algebra over $\mathbb{Z}$. It was motivated by noncommutative Witt theory; the source provides no evidence that this conjecture has been resolved.