Barbasch-Vogan characterization of unipotent representations

Let $G$ be either an orthogonal group or a metaplectic group. Let $\mathcal O_{\bold D}$ be the orbit indexing $\Pi_{\mathcal O_{\bold D}}(G)$, let $\mathcal O_{\bold d}$ be a rigid orbit, and let $\mathcal I(\pi)$ denote the infinitesimal character. Barbasch-Vogan shortest-infinitesimal-character conjecture. If $\pi\in\Pi_{\mathcal O_{\bold D}}(G)$ and $\lambda$ is the shortest infinitesimal character among all $\mathcal I(\pi)$ with $\pi\in\Pi_{\mathcal O_{\bold D}}(G)$, then

$$\lambda=\mathcal I(\bold d).$$

The source explicitly says this conjecture is false when $\mathcal O_{\bold d}$ is not rigid; under the stated rigidity hypothesis its resolution is not supplied.