The stronger Kac conjecture for Liouville quantum gravity surfaces

Let $(\boldsymbol{\lambda}_n)_{n\ge 0}$ be the eigenvalue sequence of Liouville Brownian motion on a random surface $(D,h)$, where $D$ is the domain and $h$ is the Gaussian free field. Stronger Kac conjecture. The sequence $(\boldsymbol{\lambda}_n)_{n\ge 0}$ determines the pair $(D,h)$ modulo equivalence of random surfaces. This strengthens the fixed-domain formulation by allowing the spectrum to determine the domain as well as the field, up to the natural equivalence of random surfaces.