The Steinberg representation conjecture for real reductive groups

Let $G$ be a real reductive group, and let $St_G$ denote the Steinberg representation of $G({\mathbb R})$. Let $D_2: W_{\mathbb R}\rightarrow \operatorname{SL}_2({\mathbb C})$ be the Langlands parameter associated to the lowest discrete series representation of $\operatorname{PGL}_2({\mathbb R})$. Let $\imath_G: \operatorname{SL}_2({\mathbb C})\rightarrow {}^L G$ be the Jacobson–Morozov map associated to a regular unipotent element in $\widehat{G}$. Steinberg representation conjecture. The representation $St_G$ is a direct sum of irreducible tempered representations forming a full $L$-packet on $G({\mathbb R})$, and its Langlands parameter is

$$\imath_G\circ D_2: W_{\mathbb R}\rightarrow {}^L G.$$

This conjecture proposes the real-group analogue of the description of the Steinberg representation by the principal $\operatorname{SL}_2({\mathbb C})$ for non-archimedean groups. The source gives no resolution of the conjecture.