The intrinsic description conjecture for the Fock space on nilpotent orbits

Let $\mathcal{X}_\lambda$ be the nilpotent orbit associated with the parameter $\lambda$, let $\mathcal{O}(\mathcal{X}_\lambda)$ denote its holomorphic functions, and let $\mathcal{F}_\lambda$ be the corresponding Fock space. Let $\omega_\lambda$ be the weight function and $\mathrm{d}\nu_\lambda$ the measure on $\mathcal{X}_\lambda$. Intrinsic description conjecture. The Fock space is precisely the space of square-integrable holomorphic functions:

$$\mathcal{F}_\lambda = \left\{F\in\mathcal{O}(\mathcal{X}_\lambda):\int_{\mathcal{X}_\lambda}{|F(z)|^2\omega_\lambda(z)\,\mathrm{d}\nu_\lambda(z)}<\infty\right\}.$$

This gives an intrinsic description of $\mathcal{F}_\lambda$ in terms of holomorphicity and weighted square-integrability. The conjecture was proved in the cited work for the minimal orbit, namely for $\lambda\in\mathcal{W}$ with $\mathcal{X}_\lambda=\mathcal{X}_1$, while the general case is not resolved in the supplied text.