Local convergence under conditioning of large total mass

Let $X$ be the canonical process under the excursion measure $\mathbf{N}$ of the strong Markov process $X-I$ at zero. Write $H$ for the height process, $\zeta$ for its lifetime, and $\sigma$ for the total mass of the Lévy tree. Let $\overleftarrow{H}'$ and $\overrightarrow{H}'$ denote the height processes defining the continuum condensation tree. Assume that $\alpha>0$ and that, for every $a>0$,

$$\int_1^\infty e^{a\theta}\,\pi(d\theta)=\infty.$$

Large-total-mass local convergence conjecture. As $r\rightarrow\infty$,

$$(H_{t\wedge \zeta},H_{(\zeta-t)_+};t\geq 0) \text{ under }\mathbf{N}[\cdot\mid\sigma > r]\longrightarrow (\overleftarrow{H}'_t,\overrightarrow{H}'_t;t\geq 0)$$

weakly in $\mathbb{C}([0,\infty),\mathbb{R}^2)$. This conjectures that subcritical Lévy trees conditioned on large total mass converge locally to continuum condensation trees, extending the analogous Galton–Watson result under a sufficiently heavy-tailed offspring distribution; the statement is presented as an open problem.