Unipotent representation and ideal conjecture for real reductive groups

Let $G_{\mathbb R}$ be a real reductive group, let $\widetilde{\mathcal O}$ be a birationally rigid nilpotent cover for its complexification, and let $\mathcal O_{\theta}$ be a $K$-orbit on $\mathcal O\cap\mathfrak p^*$. Write $\mathrm{Unip}_{(\widetilde{\mathcal O},\mathcal O_{\theta})}(G_{\mathbb R})$ for the finite set of equivalence classes of unipotent representations attached to this pair, and $I(\widetilde{\mathcal O})$ for the corresponding unipotent ideal. Unipotent representation and ideal conjecture. For every such pair, the constituents of $\mathrm{Unip}_{(\widetilde{\mathcal O},\mathcal O_{\theta})}(G_{\mathbb R})$ are unitary, and the ideals $I(\widetilde{\mathcal O})$ appearing in the construction are weakly unipotent. The conjecture supplies the unitarity and ideal-theoretic properties needed for the proposed orbit-method construction; its validity in the stated generality is not established in the source.