Limit of Brownian trees with exponential weight on its height

We consider a Brownian continuum random tree $\tau$ and its local time process at level $s$, say $Z_s$, which evolves as a Feller branching diffusion. Denote by $H(\tau)$ and ${\mathbb N}$ the height and the law of the tree $\tau$, respectively. Let $\mu\in {\mathbb R}$ be a constant. We show that under
$$ \frac{{\mathbb N}\left[ {\rm e}^{-\mu H(\tau)}(\tau, Z)\in \cdot\bigg{|}\int_0^{\infty}Z_s{\rm d}s=r\right]} {{\mathbb N}\left[ {\rm e}^{-\mu H(\tau)}\bigg{|}\int_0^{\infty}Z_s{\rm d}s=r\right] } \overset{d}{\longrightarrow} \text{Law}\left[ (\tau^{\mu}, Z^{\mu})\right],\quad \text{in a local sense,} $$
where if $\mu<0$, then $\tau^{\mu}$ is a Kesten tree and if $\mu>0$, then $\tau^{\mu}$ is the so-called Poisson tree constructed in Abraham, Delmas and He (2022, arXiv) by studying the local limits of $\tau$. Moreover, $Z^\mu$ is the local time process of $\tau^\mu$, which is a new diffusion, as already proved by Overbeck in 1994 by studying the Martin boundary of $Z$. We give a new representation of this diffusion using an elementary SDE with a Poisson immigration. The talk is based on some ongoing works with Romain Abraham, Jean-François Delmas and Meltem Ünel.