The ratio inequality for expected exit times under symmetrization

Let $U$ and $R$ be domains in the setting of Theorem~, let $z$ be the specified starting point, and write $\tau_U$, $\tau_R$, $\tau_{U^+}$, and $\tau_{R^+}$ for the corresponding first exit times. Ratio conjecture.

$$\frac{E^z\left(\tau_{U^+}\right)}{E^z\left(\tau_{R^+}\right)} \leq \frac{E^z\left(\tau_{U}\right)}{E^z\left(\tau_{R}\right)}.$$

This conjecture is motivated by the conditional-expectation inequality proved immediately beforehand and by related ratio inequalities in the cited literature; its resolution is not specified in the source.