Conjecture on the distributional pointwise heat-kernel asymptotics of Liouville Brownian motion

Let $\textbackslash \varphi$ be the Gaussian free field defining Liouville quantum gravity, let $\textbackslash mu_\textbackslash \varphi$ be the associated Liouville measure, and sample $x$ from $\textbackslash mu_\textbackslash \varphi$. Let $\textbackslash p_t(x,x)$ denote the diagonal heat kernel of Liouville Brownian motion. Heat-kernel asymptotics conjecture. As $t\textbackslash to0$,

$$t\textbackslash p_t(x,x)\textbackslash \Rightarrow X,$$

where $X$ is a nontrivial random variable. The conjecture is motivated by annealed Laplace-transform asymptotics, but the stated convergence in distribution remains unproved.