Conjecture on the structure of nearly holomorphic automorphic forms on $\mathrm{SL}_2$

Let $G=\mathrm{SL}_2$, let $A$ be the relevant split torus, let $M$ be a Levi subgroup, and let $\tau$ be the representation used to define the induced automorphic data. Let $\mathcal{N}(A \backslash G)_{(M,\tau)}$ denote the corresponding $(M,\tau)$-isotypic subspace of nearly holomorphic automorphic forms, let $\mathcal{E}_0(M,\tau)$ be the space spanned by the regularized Eisenstein series constructed from $\mathcal{E}(M,\tau)$, and let $\mathcal{Z}$ be the relevant center acting on $\mathcal{N}(A \backslash G)$.

The conjecture. (1) $\mathcal{N}(A \backslash G)_{(M,\tau)}\subset \mathcal{E}_0(M,\tau)$. (2) The action of $\mathcal{Z}$ on $\mathcal{N}(A \backslash G)$ is semisimple. (3) If $\mathcal{N}(A \backslash G)_{(M,\tau)}\neq 0$, then the infinitesimal character of $\tau$ is integral.

This conjecture proposes that the nearly holomorphic automorphic forms in each $(M,\tau)$-component are generated by the constructed Eisenstein series, with semisimple center action and an integrality constraint on the infinitesimal character. The supplied text does not state whether these assertions have been proved or disproved.