Conjecture on fluctuations of singular derivative Gibbs functionals

Let $Z_t(F)$ be the functional of the derivative Gibbs measure associated with branching Brownian motion, and suppose that $F(x)\sim x^{-\alpha}$ as $x\to0$ for some $\alpha$ satisfying $2<\alpha<3$. Fluctuation-order conjecture. The fluctuations of $Z_t(F)$ are of order $t^{\beta}$ for some $\beta$ satisfying

$$-\frac12<\beta<0.$$

The conjecture concerns the regime between the established $t^{-1/2}$ fluctuation scale for $\alpha<2$ and the expected order-one fluctuations when $\alpha>3$, where the functional is supported by extremal particles. The appropriate fluctuation exponent in the intermediate regime remains open.