Horospherical Cauchy transform conjecture for holomorphic discrete-series parameters

Let ${\Bbb D}(X)$ be the algebra of $G$-invariant differential operators on $X$, viewed as holomorphic differential operators on $X_\mathbb{C}$. Let $\mathcal{O}(D_+)_\lambda$ denote the common holomorphic ${\Bbb D}(X)$-eigenfunctions on $D_+$ with infinitesimal character $\lambda-\rho$. For a holomorphic function on $\Xi_+$, say it is bounded away from the boundary if its restriction to $g\mathcal{T}_+a$ is bounded for every $g\in G$ and $a\in A_+$; write $\mathcal{O}_{\rm b.a.b.}(\Xi_+)$ for the space of such functions, and analogously write $\mathcal{O}_{\rm b.a.b.}(D_+)_\lambda$ for the corresponding eigenspace on $D_+$. Let $\lambda\in\Lambda_2$. Horospherical Cauchy transform conjecture.

$$\mathcal{O}_{\rm b.a.b.}(D_+)_\lambda=L^2(X)_\lambda^{-\omega}.$$

The inclusion “$\supset$” is clear, and the conjecture would imply that the horospherical Cauchy transform induces an intertwining isomorphism

$$\mathcal{O}_{\rm b.a.b.}(D_+)_\lambda\to\mathcal{O}(\Xi_+)_\lambda.$$

This is a holomorphic analogue of Helgason's conjecture and is intended here for parameters $\lambda\in\Lambda_2$; formulating and proving the analogous statement for other parameters remains open.