Consistency of the Boolean Prime Ideal Theorem with no Vitali set

The Boolean Prime Ideal Theorem (BPI) asserts that every Boolean algebra has an ultrafilter. A Vitali set is a subset of the real line containing exactly one representative from each equivalence class modulo rational differences.

Consistency conjecture. It is consistent with ZF plus the Axiom of Dependent Choices (DC) that BPI holds and there is no Vitali set in the real line.

This conjecture concerns the relationship between choice principles in choiceless set theory. BPI does not imply the Axiom of Choice in ZF, while it remains open whether BPI is equivalent to AC over ZF+DC; the proposed consistency statement would separate BPI from the existence of Vitali sets in that setting.