We prove that Späth's Character Triple Conjecture holds for every finite group with respect to maximal defect characters at the prime 2. This is done by reducing the maximal defect case of the conjecture to the so-called inductive Alperin–McKay condition whose verification has recently been completed by Ruhstorfer for the prime 2. As a consequence we obtain the Character Triple Conjecture for all 2-blocks with abelian defect groups by applying Brauer's Height Zero Conjecture, a proof of which is now available. We also obtain similar results for the block-free version of the Character Triple Conjecture at the prime 3.
