Non-isomorphism conjecture for cyclotomic restricted rational Cherednik algebras

Let $m$ be even, let $\frac{m}{p}$ be odd, and set $k=\mathbb{Q}(\sqrt[m]{1})$. Let $\underline{\overline{H}}_{\underline{c}}(\mu(G(m,p,n)))$ and $\overline H_c(G(m,p,n))$ be the two $k$-algebras defined in the paper. Non-isomorphism conjecture. The $k$-algebras

$$\underline{\overline{H}}_{\underline{c}}(\mu(G(m,p,n)))\quad\text{and}\quad \overline H_c(G(m,p,n))$$

are not isomorphic. The conjecture is motivated by an explicit example showing non-isomorphism over $\mathbb{Q}$; the authors note that this example becomes isomorphic after extending scalars to $\mathbb{Q}(\sqrt{-1})$, and that they have no explicit example disproving isomorphism over $\mathbb{C}$.