Characterization conjecture for weighted composition operators on Bloch spaces

Let $D$ be a bounded homogeneous domain in $\mathbb{C}^n$, let $\psi$ be a holomorphic function on $D$, and let $\varphi$ be a holomorphic self-map of $D$. Write $W_{\psi,\varphi}f=\psi(f\circ\varphi)$ for the weighted composition operator, and let $\mathcal{B}(D)$ denote the Bloch space of $D$. The quantities $\sigma_{\psi,\varphi}$ and $\tau_{\psi,\varphi}$ are the associated finiteness conditions defined in the paper. Weighted composition operator conjecture. The operator $W_{\psi,\varphi}$ is bounded on $\mathcal{B}(D)$ if and only if $\psi\in\mathcal{B}(D)$ and $\sigma_{\psi,\varphi}$ and $\tau_{\psi,\varphi}$ are finite. Furthermore, the bounded operator $W_{\psi,\varphi}$ is compact on $\mathcal{B}(D)$ if and only if

$$\lim_{\varphi(z)\to\partial D}\omega(\varphi(z))Q_\psi(z)=\lim_{\varphi(z)\to\partial D}|\psi(z)|T_\varphi(z)=0.$$

The theorem immediately preceding the conjecture proves sufficiency of the compactness conditions, while the analogous necessity is known for the unit ball and polydisk but not for general bounded homogeneous domains. The conjecture also asserts the corresponding necessary and sufficient boundedness characterization.