Local convergence under conditioning of large maximal degree

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 $\Delta X$ for the jumps of $X$. Let $\overleftarrow{H}'$ and $\overrightarrow{H}'$ denote the height processes defining the continuum condensation tree. Assume that $\alpha>0$ and that the Lévy measure $\pi$ has unbounded support. The local convergence conjecture. As $r\rightarrow\infty$,

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

weakly in $\mathbb{C}([0,\infty),\mathbb{R}^2)$. This is the expected local limit of subcritical Lévy trees conditioned to have large maximal degree, analogous to known local-convergence results for Galton–Watson trees conditioned on large maximal out-degree and continuous-state branching processes conditioned on a large maximal jump; the proof is not currently available.