In supercritical branching Brownian motion, the derivative martingale converges to a nontrivial limit that plays a central role in describing the extremal process and the Gibbs measures near the frontier of the cloud. The small value probability, meaning the probability that this limit falls below a small positive threshold epsilon, is expected to decay like a power of epsilon as epsilon goes to zero. The open question is whether this power-law exponent is universal across all supercritical branching mechanisms satisfying standard moment conditions, or whether it depends sensitively on the fine details of the offspring distribution. Current results establish the asymptotic form in specific cases such as binary branching, but a general characterization across the full class of supercritical offspring distributions remains unresolved. The difficulty stems from the fact that the derivative martingale limit is not a simple functional of the branching mechanism; it encodes intricate correlations between the spatial positions of particles at all generations, and its distributional tail is shaped by rare large families that contribute in ways that differ subtly depending on the offspring law. Existing techniques based on spine decompositions and many-to-one lemmas give sharp estimates in binary or Poisson cases but break down when higher-order cumulants of the offspring distribution are significant, because the correction terms in the asymptotic expansion become entangled in a way that resists standard comparison arguments. Resolving this universality question would clarify the structure of the Liouville quantum gravity measures that arise as scaling limits, settle the universality class of log-correlated Gaussian fields in branching models, and provide new tools for the stochastic Keller-Segel system where analogous martingale quantities control the onset of blow-up in the critical regime.