Semisimplicity conjecture for the Cohen–Wales algebra

Let $l$, $m$ and $r$ be three nonzero complex numbers with $m=\frac{1}{r}-r$. The Cohen–Wales algebra (CGW algebra) with parameters $l$ and $m$ is considered over the field $\mathbb{Q}(l,r)$.

Semisimplicity conjecture. The CGW algebra is semisimple except possibly if $r$ is a root of unity or if

$$l\in\bigcup_{k=4}^n\{r^{2k-5},-r^{5-2k},-r^{4k-7},r^{7-4k},r^{7-2k},-r^{2k-7}\}.$$

This conjecture gives the expected exceptional parameter values for semisimplicity, in the spirit of known semisimplicity criteria for related algebras. The source provides no resolution, so the conjecture is recorded as open.