A discriminated problem about discriminants

Fairly optimal lower bounds are known for the discriminants of number fields are known but they seem to be far from the truth in the case of monogenic number fields, that is, for the discriminants of (monic) irreducible polynomials with integer coefficients. This problem appears in the 1990 review of Odlyzko as Open Problem 2.5, with attribution to Serre "and others." I will try to update on the recent development and connections with Lehmer's question on the Mahler measure.