  We make progress on the weak-universality conjecture for the KPZ equation by
proving the height functions associated to a class of non-integrable and
non-stationary generalizations of ASEP converge to the solution of the
appropriate KPZ stochastic PDE. The models considered herein are variations on
ASEP with a mesoscopic family of slow bonds, thus our results partially extend
those of [Franco-Goncalves-Simon '16] to the non-stationary regime and add to
the almost empty set of non-integrable and non-stationary interacting particle
systems for which weak KPZ universality has been established. Technically, in
this current paper we develop further the dynamical strategy from [Y '20] and
[Y '20b] through spatial localization and combination with the two-blocks
scheme of [Guo-Papanicolaou-Varadhan '88] and [Dembo-Tsai '16] with hopes
towards weak KPZ universality in more generality.
