Local limit conjecture for conditioned sub-critical non-generic marked Galton–Watson trees

Let $\tau^{*}$ be a sub-critical and non-generic marked Galton–Watson tree with offspring distribution $\mathbf{p}$ satisfying the condition in

, with $\rho_l(\mathbf{p},\mathbf{q})=1$, and mark function $\mathbf{q}$ satisfying the condition in

. Let $M(\tau^*)$ denote the total mark and let $\tau_C^{*}(\mathbf{p},\mathbf{q})$ be the corresponding condensation-tree limit. Local limit conjecture. The conditioned tree converges in distribution as the total mark tends to infinity:

$$\mathrm{dist}(\tau^{*}\mid M(\tau^*)=n)\underset{n\to+\infty}{\longrightarrow} \mathrm{dist}(\tau_C^{*}(\mathbf{p},\mathbf{q})).$$

The conjecture proposes the local limit in the sub-critical, non-generic case not covered by the paper's results. Its validity is left open in the supplied text.