Dimension-free linear weighted bound for the Cauchy transform

Let $w\in A_2$ be a weight on the unit sphere in $\mathbb C^n$, let $C$ denote the Cauchy transform in $\mathbb C^n$, and let $\widetilde Q_2(w)$ denote the invariant $A_2$ characteristic of $w$. The notation $L^2(w)$ denotes the corresponding weighted $L^2$ space. The question is whether

$$\|C(f)\|_{L^2(w)}\lesssim \widetilde Q_2(w)\|f\|_{L^2(w)},$$

with an implied constant independent of the dimension.

Dimension-free Cauchy-transform conjecture. For every $w\in A_2$ and the Cauchy transform $C$ in $\mathbb C^n$, the displayed estimate holds with an implied constant independent of the dimension.

A linear estimate of this form is described as the goal in the concluding remarks, and the source says that the required result is not known. The paper's square-function estimate supports the proposed dependence on the invariant characteristic, but the Cauchy-transform estimate remains open because the needed several-variable gradient identity fails.