Frobenius semisimplicity and Heegner–Drinfeld classes conjecture

Let $r=\operatorname{ord}_{s=1/2}L(\pi_{F'},s)$ and let $\mu\in\mathfrak{T}_{r,\Sigma}$. The class $Z_\pi^\mu(\xi)$ lies in

$$\pi^K\otimes\wedge^r\left(W'^{\operatorname{Fr}=q}_\pi\right)\otimes\ell_{\pi,\xi}.$$

Frobenius semisimplicity conjecture. For the eigenvalue $q$, the generalized eigenspace of the $\operatorname{Fr}$-action on $W'_\pi$ coincides with the eigenspace, and $Z_\pi^\mu(\xi)$ gives a basis of the line

$$\pi^K\otimes\wedge^r\left(W'^{\operatorname{Fr}=q}_\pi\right)\otimes\ell_{\pi,\xi}.$$

The conjecture is motivated by the standard conjecture on Frobenius semisimplicity. The cited result establishes nonvanishing of $Z_\pi^\mu(\xi)$ under the stated order-of-vanishing condition, but the asserted identification of generalized and ordinary eigenspaces remains unproved in the supplied text.