Lacoin–Rhodes–Vargas left-tail conjecture for derivative Gaussian multiplicative chaos

Let $D\subset\mathbb{R}^d$ and let a log-correlated Gaussian field on $D$ admit a smooth white noise decomposition. Denote its derivative Gaussian multiplicative chaos by $\mathcal{D}_\infty(\mkern 1.5mu\overline{\mkern-1.5mu\gamma\mkern-1.5mu}\mkern 1.5mu)$, and let $\mkern 1.5mu\overline{\mkern-1.5mu\gamma\mkern-1.5mu}\mkern 1.5mu_c$ be the critical parameter. Lacoin–Rhodes–Vargas' left-tail conjecture. For every $\mkern 1.5mu\overline{\mkern-1.5mu\gamma\mkern-1.5mu}\mkern 1.5mu\in(0,\mkern 1.5mu\overline{\mkern-1.5mu\gamma\mkern-1.5mu}\mkern 1.5mu_c)$,

$$-\log \mathbf{P}(\mathcal{D}_\infty(\mkern 1.5mu\overline{\mkern-1.5mu\gamma\mkern-1.5mu}\mkern 1.5mu)<-v)\asymp v^{(\mkern 1.5mu\overline{\mkern-1.5mu\gamma\mkern-1.5mu}\mkern 1.5mu_c/\mkern 1.5mu\overline{\mkern-1.5mu\gamma\mkern-1.5mu}\mkern 1.5mu)^2}.$$

This conjecture sharpens the previously known sub-Gaussian left-tail bound in the $L^4$ regime $\mkern 1.5mu\overline{\mkern-1.5mu\gamma\mkern-1.5mu}\mkern 1.5mu\in(0,\mkern 1.5mu\overline{\mkern-1.5mu\gamma\mkern-1.5mu}\mkern 1.5mu_c/2)$ and predicts the left-tail exponent throughout the entire subcritical regime. The supplied source does not state that the conjecture has been resolved.