Galvin's Conjecture and Weakly Precipitous Ideals

We investigate a combinatorial game on $\omega_1$ and show that mild large cardinal assumptions imply that every normal ideal on $\omega_1$ satisfies a weak version of precipitousness. As an application, we show that that the Raghavan-Todor\v{c}evi\'{c} proof of a longstanding conjecture of Galvin (done assuming the existence of a Woodin cardinal) can be pushed through under much weaker large cardinal assumptions.