We show that every two-way deterministic finite automaton (2dfa) that solves one-way liveness on height h has \(\varOmega (h^2)\)  states. This implies a quadratic lower bound for converting one-way nondeterministic finite automata to 2dfas, which asymptotically matches Chrobak’s well-known lower bound for this conversion on unary languages. In contrast to Chrobak’s simple proof, which relies on a 2dfa’s inability to differentiate between any two sufficiently distant locations in a unary input, our argument works on alphabets of arbitrary size and is structured around a main lemma that is general enough to potentially be reused elsewhere.