  We use the RG framework set up in arXiv:2009.10087 to explore the $\phi^3$
theory with a random field interaction. According to the Parisi-Sourlas
conjecture this theory admits a fixed point with emergent supersymmetry which
is related to the pure Lee-Yang CFT in two less dimensions. We study the model
using replica trick and Cardy variables in $d=8-\epsilon$ where the RG flow is
perturbative. Allowed perturbations are singlets under the $S_n$ symmetry that
permutes the $n$ replicas. These are decomposed into operators with different
scaling dimensions: the lowest dimensional part, `leader', controls the RG flow
in the IR; the other operators, `followers', can be neglected. The leaders are
classified into: susy-writable, susy-null and non-susy-writable according to
their mixing properties. We construct low lying leaders and compute the
anomalous dimensions of a number of them. We argue that there is no operator
that can destabilize the SUSY RG flow in $d\le 8$. This agrees with the well
known numerical result for critical exponents of Branched Polymers (which are
in the same universality class as the random field $\phi^3$ model) that match
the ones of the pure Lee-Yang fixed point according to dimensional reduction in
all $2\le d\le 8$. Hence this is a second strong check of the RG framework that
was previously shown to correctly predict loss of dimensional reduction in
random field Ising model.
