Diameter estimates in Kähler geometry II: removing the small degeneracy assumption

In this short note, we remove the small degeneracy assumption in our earlier works [10, 11]. This is achieved by a technical improvement of Corollary 5.1 in [10]. As a consequence, we establish the same geometric estimates for diameter, Green’s functions and Sobolev inequalities under an entropy bound for the Kähler metrics, without any small degeneracy assumption.