Kostant's canonical extension conjecture for minimal K-types

Let $G_0$ be a real reductive linear Lie group, and let $K_0$ be its maximal compact subgroup. Fix an Iwasawa decomposition

$$G_0=K_0A_0N_0$$

of $G_0$, where $A_0$ is abelian and normalizes $N_0$. Let $G$, $K$, and $A$ be the complexifications of $G_0$, $K_0$, and $A_0$, respectively, and let $e\in G/K$ correspond to $[K]\in G/K$. Let $V$ be an irreducible $K$-module occurring as the minimal $K_0$-type of a unitary representation of $G_0$. Let $Y$ be the wonderful compactification of $G/K$. Kostant's conjecture. There exists a $G$-equivariant vector bundle ${\mathcal E}_V$ on $Y$ such that:

1. ${\mathcal E}_V|_{G/K}\cong G\times_KV$ as $G$-equivariant vector bundles;
2. for every $v\in V\cong {\mathcal E}_V\otimes_{{\mathcal O}_Y}k(e)$ and every one-parameter subgroup $\tau:\mathbb{G}_m\to A$, the limit $\lim_{t\to\infty}\tau(t)v$ exists in the total space $V({\mathcal E}_V)$ of ${\mathcal E}_V$;
3. for every $G$-equivariant vector bundle ${\mathcal E}$ on $Y$ satisfying the preceding two properties, there is a $G$-equivariant embedding ${\mathcal E}\hookrightarrow {\mathcal E}_V$.

Kostant's question seeks a canonical algebraic extension encoding the boundary behavior of equivariant vector bundles associated with minimal $K$-types, in connection with the asymptotic behavior of minimal $K$-types and Casselman's theorem on $\mathfrak{n}_0$-coinvariants. The source gives no resolution of the conjecture, so its status is open.