As noted or hinted upon by others, the concepts challenged in this paper (i.e. general relativity) are very well established and tested experimentally, e.g., in the context of GPS. Conversely, the question of what negative masses or mass densities would imply is obvious, so that probably every single physics student or professor has thought about it. Still, nobody seems to have found any traces of negative masses.
Thus, the authors suggestions should be viewed as (potentially) somewhat interesting speculation or an empirical approach for fitting cosmological data.
I dont think there is anything new that the general public could/should learn from this news or the article.