Silicon and oxygen are potential light elements in Earth's core because their stronger affinity to metal observed with increasing temperature posits that significant amounts of both can be incorporated into the core. It was proposed that an Fe-Si-O liquid alloy could expel SiO2 at the core-mantle boundary during secular cooling, leaving the core with either silicon or oxygen, not both. This was recently challenged in a study showing no exsolution but immiscibility in the Fe-Si-O system. Here we investigate the liquidus field of Fe-Si and Fe-O binaries and Fe-Si-O ternaries at core-mantle boundary pressures and temperatures using ab initio molecular dynamics. We find that the liquids remain well mixed with ternary properties identical to mixing of binary properties. Two-phase simulations of solid SiO2 and liquid Fe show dissolution at temperatures above 4100 K, suggesting that SiO2 crystallization as well as liquid immiscibility in Fe-Si-O is unlikely to occur in Earth's core.