The Kac conjecture for Liouville quantum gravity

Let $D$ be a fixed bounded domain, let $h$ be the Gaussian free field on $D$, and let $(\boldsymbol{\lambda}_n)_{n\ge 0}$ be the eigenvalues of Liouville Brownian motion associated with the Liouville measure generated by $h$. Kac's conjecture. One can almost surely hear the shape of Liouville quantum gravity: there exists a measurable function $\phi$ such that

$$h=\phi\bigl((\boldsymbol{\lambda}_n)_{n\ge 0}\bigr),$$

almost surely. The domain is fixed and known; the conjecture asks whether the spectrum determines the underlying Gaussian free field. A stronger version determines $(D,h)$ modulo equivalence of random surfaces.