Remarks on Gross' technique for obtaining a conformal Skorohod embedding of planar Brownian motion

In a recent work by Gross, it was proved that, given a distribution $\mu$ with zero mean and finite second moment, we can find a simply connected domain $\mathrm{\Omega }$ such that if $Z_t$ is a standard planar BM, then $\mathcal{R}e(Z_{{\tau }_{\mathrm{\Omega }}})$ has the distribution $\mu$. In this note, we extend his method to prove that if $\mu$ has a finite $L^p$ moment then the exit time ${\tau }_{\mathrm{\Omega }}$ has a finite moment of order $\frac{p}{2}$.