Decay of correlations conjecture for the massless hierarchical Liouville model

Let $\nu_{L,m,\epsilon}^\mathrm{Lv}$ be the measure defined by the discrete Gaussian free field Liouville density, and let $\phi\sim\nu_{L,m,\epsilon}^\mathrm{Lv}$. For $f\in C_c^\infty(\mathbb R^2)$ and $x\in\mathbb R^2$, write $f_x(\cdot)=f(\cdot-x)$. There is a family of events $(\mathcal A_{x,r,L})_{x\in\mathbb R^2,\,r\geq0,\,L>0}$ such that

Decay of correlations conjecture. The events satisfy $\nu_{L,m,\epsilon}^\mathrm{Lv}(\mathcal A_{x,r,L})\to1$ as first $\epsilon\to0$, then $m\to0$, and finally $L\to\infty$, and, for every $f,g\in C_c^\infty(\mathbb R^2)$,

$$\limsup_{r\to\infty}\limsup_{L\to\infty}\limsup_{m\to0}\limsup_{\epsilon\to0}\sup_{y\in\mathbb R^2\colon |x-y|=r}\left|\left\langle\phi(f_x)\phi(g_y)\mathbf 1_{\mathcal A_{x,r,L}}\right\rangle_{\nu_{L,m,\epsilon}^\mathrm{Lv}}\right|=0.$$

This conjectures decay of correlations for the massless hierarchical Liouville model in infinite volume after removing the ultraviolet cutoff, the mass, and then the finite-volume scale. The events allow the correlation estimate to be restricted to a high-probability set under the regularized Liouville measure; the source provides no resolution of the conjecture.