Saito–Kurokawa type lift conjecture for quaternionic Hecke eigenforms

Let $h$ be a cuspidal Hecke eigenform on $\mathbf{SL}_{2}(\mathbb{Z})$ of weight $2\ell-n+2$, and let $F_h$ be the function on $\mathbf{G}(\mathbb{A})$ whose constant term along $Z$ has Fourier expansion

$$F_{h,Z}(g)=\sum_{\substack{T\in \mathbf{V}_{0}\\ \langle T,T\rangle>0}} A_h(T)(g_f)\mathcal{W}_{T}(g_\infty).$$

Here $A_h(T)(g_f)$ is defined using the Satake parameters of the automorphic representation associated with $h$ as in the preceding construction. Saito–Kurokawa type lift conjecture. The function $F_h$ is a quaternionic cuspidal Hecke eigenform of weight $\ell$ on $\mathbf{G}$. This conjecture predicts that the Fourier expansion constructed from the cuspidal eigenform $h$ produces a genuine quaternionic cuspidal Hecke eigenform on $\mathbf{G}$; the supplied text gives no evidence that the conjecture has been resolved.