Singularities of divisors of low degree on simple abelian varieties

It is known by results of Koll'ar, Ein, Lazarsfeld, Hacon and Debarre that divisors representing principal and other low degree polarizations on abelian varieties have mild singularities. In this note we extend such results to polarizations of degree $<g$ on simple $g$-dimensional abelian varieties, settling a conjecture of Debarre and Hacon.