  We prove here that the height function associated to non-simple exclusion
processes with arbitrary jump-length converges to the solution of the
Kardar-Parisi-Zhang SPDE under suitable scaling and renormalization. This
extends the work of Dembo-Tsai'16 for arbitrary jump-length and Goncalves-Jara
'17 for the non-stationary regime. Thus we answer a ``Big Picture Question"
from the AIM workshop on KPZ and also expand on the almost empty set of
non-integrable and non-stationary particle systems for which weak KPZ
universality is proven. We use an approximate microscopic Cole-Hopf transform
as in Dembo-Tsai'16 but we develop tools to analyze local statistics of the
particle system via local equilibrium and work of Goncalves-Jara '17. Local
equilibrium is done via the one-block step in Guo-Papanicolaou-Varadhan '88 for
path-space/dynamic statistics.
