A two-sided local-limit conjecture for the diffusion approximation of tree profiles

For $i=0,1$, let $(\delta_i,m_i,s_i)\in\mathbb{R}\times(0,+\infty)\times[0,1]$, and let $(\delta_i^n,m_i^n,s_i^n)$ be a sequence in $\mathbb{Z}^3$ with $m_i^n>0$. Set

$$z_i^n:=\left(\delta_i^n/n^{1/2},m_i^n/n^{3/4},s_i^n/n\right)\to z_i:=(\delta_i,m_i,s_i),\qquad i\in\{0,1\}.$$

Let $0<\epsilon<s_1$, and let $\xi^\star(t)$ be the process from the diffusion approximation theorem, stopped when $m^\star$ hits $\epsilon$ and started at $z_0$ at time $0$. Let $\mathsf{P}_{\geq \epsilon n^{3/4}}$ denote the discrete process $(\bar{\xi}^n)$ started at $z_0^n$ and conditioned to equal $z_1^n$ at time $t_1$.

Two-sided local-limit conjecture. The conditioned discrete process converges in distribution in $D([0,t_1],\mathbb{R}^3)$ to $\xi^\star$, started at $(\delta_0,m_0,s_0)$ and conditioned by $\xi^\star(t_1)=z_1$.

This conjecture would provide the missing local-limit statement for the diffusion approximation under conditioning at both boundaries. The supplied status evidence indicates that the corresponding profile limit is known through convergence to Brownian-excursion local time and its stochastic differential equation, so this claim is treated as solved.