We discuss axionlike particles (ALPs) within the framework of Higgs effective field theory, targeting instances of close alignment of ALP physics with a custodial singlet character of the Higgs boson. We tension constraints arising from new contributions to Higgs boson decays against limits from high-momentum transfer processes that become under increasing control at the LHC. Going beyond leading-order approximations, we highlight the importance of multitop and multi-Higgs production for the pursuit of searches for physics beyond the Standard Model extensions.