If $X$ is a smooth (complex) projective variety of maximal Albanese dimension such that $\chi(\omega_X)>0$, how does one show that $X$ is of general type?

I've seen this used but I can't find a proof.

It is a result of Ein-Lazarsfeld that if $X$ is a smooth projective variety of maximal albanese dimension such that $\chi(\omega_X)=0$, then the albanese image is fibered by tori, so in partixular if $X$ is birational to its image under the albanese map, then it can't be of general type.