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 $\Omega$ such that if $Z_{t}$ is a standard planar BM, then $\mathcal{R}e(Z_{\tau_{\Omega}})$ 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 $\tau_{\Omega}$ has a finite moment of order $\frac{p}{2}$.