Structural conjecture for the strict endoscopic part of modular Siegel threefolds

Let $X_N$ be the modular Siegel threefold of level $N$, let $\mathbb{V}(\lambda)$ be the local system associated with the weight $\lambda$, and let

$$\mathcal{C}=\{\pi=\pi_\infty\otimes\pi_{fin}\mid \pi\text{ is irreducible and }H^{3,0}_{!}(X_N,\mathbb{V}(\lambda))(\pi_{fin})^{K(N)}=0\}.$$

The strict endoscopic part is

$$H_{\mathrm{End}^{\mathrm{s}}}^{3}(X,\mathbb{V}(\lambda))=\bigoplus_{\pi\in\mathcal{C}}m(\pi)(\pi_{fin})^{K(N)}.$$

Structural conjecture for the strict endoscopic part. The strict endoscopic part contributes as a finite part of theta-lifts of cuspidal automorphic representations of $(\operatorname{GL}(2)\times\operatorname{GL}(2))/\mathbb{G}_m$ over $\mathbb{Q}$. This conjecture proposes a structural description of the portion of inner cohomology not detected by the holomorphic part of the Hodge structure; it is attributed in the source to numerical calculations and to the authors cited there. Its resolution is not specified in the supplied text.