P-adic Gamma classes and overconvergent Frobenius structures for quantum connections

Consider the small quantum connection on a monotone symplectic manifold, with p-adic coefficients. We conjecture that this always admits an overconvergent Frobenius structure, whose constant term is given by a characteristic class associated to Morita's p-adic Gamma function. We prove this conjecture for toric Fano varieties and Grassmannians, and also supply additional experimental evidence.