Closed-range conjecture for composition operators on the Dirichlet space

Let $\mathbb D$ be the unit disk, let $\varphi$ be an analytic self-mapping of $\mathbb D$, and let $C_{\varphi}f=f\circ\varphi$ be bounded on the Dirichlet space $\mathcal D(\mathbb D)$. Write $n_{\varphi}(w)$ for the multiplicity function of $\varphi$ and $D(z,r)$ for the disk of radius $r$ centered at $z$. A measure $\mu$ is reverse Carleson when there are $\delta>0$ and $r>0$ such that $\mu(D(z,r))\geq\delta A(D(z,r))$ for every $z\in\mathbb D$. Closed-range conjecture. The range $R(C_{\varphi})$ is closed if and only if, for every $\alpha\in(0,1)$, there are $\delta>0$ and $r>0$ such that

$$\int_{\varphi(\mathbb D)\cap D(z,r)}n_{\varphi}^{\alpha}(w)\,dA(w)\geq\delta A(D(z,r))$$

for all $z\in\mathbb D$; equivalently, $n_{\varphi}^{\alpha}(w)dA(w)$ is a reverse Carleson measure. This conjecture proposes a necessary and sufficient geometric condition for the closed range of a bounded composition operator on the Dirichlet space.