Compactness characterization for weighted composition operators on bounded homogeneous domains

Let $D$ be a bounded homogeneous domain in $\mathbb{C}^n$, let $\psi\in H(D)$, and let $\varphi$ be a holomorphic self-map of $D$. Write $\mathcal{B}(D)$ for the Bloch space and $H^\infty_{\mu}(D)$ for the weighted Banach space. The weighted composition operator is defined by

$$W_{\psi,\varphi}f=\psi(f\circ\varphi).$$

Here $\omega$ denotes the Bloch growth function associated with $D$, and $\varphi(z)\to\partial D$ means that the image approaches the boundary of $D$. Compactness conjecture. The operator

$$W_{\psi,\varphi}:\mathcal{B}(D)\to H^\infty_{\mu}(D)$$

is compact if and only if $\psi\in H^\infty_{\mu}(D)$ and

$$\lim_{\varphi(z)\to\partial D}\frac{1}{2}\mu(z)|\psi(z)|\omega(\varphi(z))=0.$$

The sufficiency of these conditions is established in the paper. The conjectural part is their necessity, for which more information about the Bloch growth function $\omega$ on bounded homogeneous domains is needed; analogous results are known for such operators on the Bloch space in related settings.