Abstract Lyons and Sullivan conjectured in (J. Differential Geom. 19 (2), 299–323, 1984) that if p : M → N is a normal Riemannian covering, with closed, has exponential volume growth, then there are non-constant, positive harmonic functions on . This was proved recently Polymerakis (Adv. Math. 379 , 107552–107558, 2021) exploiting the Lyons-Sullivan discretization some sophisticated estimates g...