We establish a rather sharp two-side estimate for the tail probability of the derivative martingale limit in a branching random walk throughout the entire subcritical regime, confirming a conjecture by Lacoin, Rhodes, and Vargas (Duke Math. J. 171 (3): 483--545, 2022. ) for the special case of log-correlated Gaussian fields on trees.
Chen et al. (Sat,) studied this question.