  We extend the study of \emph{melonic} quartic tensor models to models with
arbitrary quartic interactions. This extension requires a new version of the
loop vertex expansion using several species of intermediate fields and iterated
Cauchy-Schwarz inequalities. Borel summability is proven, uniformly as the
tensor size $N$ becomes large. Every cumulant is written as a sum of explicitly
calculated terms plus a remainder, suppressed in $1/N$. Together with the
existence of the large $N$ limit of the second cumulant, this proves that the
corresponding sequence of probability measures is uniformly bounded and obeys
the tensorial universality theorem.
