Horinaga's conjecture on nearly holomorphic automorphic forms

Let $G_n$ be the symplectic group used in the paper, let $(M,\tau)$ be a cuspidal pair, and let $\mathcal{N}(G_n)_{(M,\tau)}$ denote the space of nearly holomorphic automorphic forms with cuspidal support $(M,\tau)$. Let $\mathcal{E}_0(M,\tau)$ be the space of leading terms of the associated Eisenstein series, and let $\mathcal{Z}_n$ be the center acting on the relevant representation space.

Horinaga's conjecture. The following statements hold:

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

The conjecture predicts both an Eisenstein-series description of each cuspidal-support summand and strong algebraic control of the center action. The supplied text does not state any resolution of these assertions.