Conjectural classification of ps-idempotents by unrefined endo-parameters

Let ${\rm G}$ be a classical group, let $\mathfrak{Z}_{{\rm G},\overline{\mathbb{Z}}[1/p]}^{\operatorname{st}}$ denote the stable center, and let $\mathcal{EP}({\rm G})$ be the set of unrefined endo-parameters for ${\rm G}$. A ps-idempotent is a primitive idempotent of this stable center. Ps-idempotent classification conjecture. There is a natural bijection

$$\left\{\text{primitive idempotents of }\mathfrak{Z}_{{\rm G},\overline{\mathbb{Z}}[1/p]}^{\operatorname{st}}\right\}\leftrightarrow\mathcal{EP}({\rm G}),$$

i.e. the ps-idempotents are given by endo-parameters after forgetting their Witt data. This is intended as a description of stable blocks in terms of endo-parameters and is asserted to be uniquely compatible with local Langlands.