Local-time convergence conjecture for conditioned Galton–Watson trees

Let $\xi$, $T_{\xi}$, and $|T_{\xi}|$ be as in the conditioned Galton–Watson tree theorem, and write $\sigma^2=\operatorname{Var}\xi$. Let $H_n^{\xi}$ and $C_n^{\xi}$ denote respectively the height and contour functions of $T_{\xi}$ conditioned to have $n+1$ vertices, and let $L(H_n^{\xi};t,y)$ and $L(C_n^{\xi};t,y)$ be the corresponding continuous local-time functions defined by the formulas in the source. Let $\mathbf{e}$ be the normalized Brownian excursion and let $\eta(2\mathbf{e}/\sigma;t,y)$ denote its local-time process. Local-time convergence conjecture. Along the subsequence for which $\operatorname{\mathbb{P}}(|T_{\xi}|=n+1)>0$, as $n\to\infty$, the joint local-time processes should converge in $C([0,1]\times\mathbb{R})$ as follows:

$$\Big(n^{-1/2}L(H_n^{\xi};nt,y\sqrt n);\ t\in[0,1],\ y\in\mathbb{R}\Big)\stackrel{\mathrm{d}}{\longrightarrow}\Big(\eta(2\mathbf{e}/\sigma;t,y);\ t\in[0,1],\ y\in\mathbb{R}\Big),$$

and

$$\Big(n^{-1/2}L(C_n^{\xi};2nt,y\sqrt n);\ t\in[0,1],\ y\in\mathbb{R}\Big)\stackrel{\mathrm{d}}{\longrightarrow}\Big(2\eta(2\mathbf{e}/\sigma;t,y);\ t\in[0,1],\ y\in\mathbb{R}\Big).$$

The conjecture would provide a direct continuum analogue of the local-time convergence used in the paper's discrete approximation arguments, avoiding the intermediary study of maps. The source presents this as a hoped-for result rather than an established theorem, and does not provide a resolution.