  We revisit perturbative RG analysis in the replicated Landau-Ginzburg
description of the Random Field Ising Model near the upper critical dimension
6. Working in a field basis with manifest vicinity to a weakly-coupled
Parisi-Sourlas supersymmetric fixed point (Cardy, 1985), we look for
interactions which may destabilize the SUSY RG flow and lead to the loss of
dimensional reduction. This problem is reduced to studying the anomalous
dimensions of "leaders" -- lowest dimension parts of $S_n$-invariant
perturbations in the Cardy basis. Leader operators are classified as
non-susy-writable, susy-writable or susy-null depending on their symmetry.
Susy-writable leaders are additionally classified as belonging to superprimary
multiplets transforming in particular $\textrm{OSp}(d | 2)$ representations. We
enumerate all leaders up to 6d dimension $\Delta = 12$, and compute their
perturbative anomalous dimensions (up to two loops). We thus identify two
perturbations (with susy-null and non-susy-writable leaders) becoming relevant
below a critical dimension $d_c \approx 4.2$ - $4.7$. This supports the
scenario that the SUSY fixed point exists for all $3 < d \leq 6$, but becomes
unstable for $d < d_c$.
