Type-II seesaw of a non-holomorphic modular $A_4$ symmetry

We search for predictability of lepton masses and mixing patterns of type-II seesaw scenario in a non-holomorphic modular $A_4$ symmetry recently proposed by Qu and Ding. We propose three types of minimum predictive models with different assignments of modular weight, satisfying the neutrino oscillation data in Nufit 5.2. The cosmological bound on the sum of neutrino mass is stringent to our models and CMB bound $\sum D_{\nu }\le 0.12$ eV can be satisfied by one of three models playing an important role in discriminating them.