Conjecture on stable fluctuations of the minimum in branching Brownian motion

Let $M_t=\min_{u\in\mathcal{N}(t)}X_u(t)$ be the minimum position in branching Brownian motion, let $Z_\infty$ be the derivative-martingale limit, and let $G$ be a standard Gumbel random variable. Let $(S_t)_{t\geq0}$ be a Cauchy process, and assume that $Z_\infty$, $G$, and $(S_t)_{t\geq0}$ are independent.

Conjecture on the minimum. As $t\to\infty$, for some constant $C>0$,

$$M_t\stackrel{\mathrm{law}}{=}\frac{3}{2}\log t-\log(CZ_\infty)-G+\frac{1}{\sqrt{t}}S_{Z_\infty}+o\mathopen{}\mathclose\bgroup\originalleft(\frac{1}{\sqrt{t}}\aftergroup\egroup\originalright).$$

This predicts a refined fluctuation expansion for the minimum beyond its usual logarithmic order. The authors state that a formal formulation should use mod-$\phi$-convergence; the conjecture is presented as a direction for future work and remains open in the supplied source.