Uniform height-tail estimate for Boltzmann triangulations

Let $\mathbf{t}$ be a Boltzmann triangulation with a boundary of perimeter $p\geq 1$, let $\mathsf{Height}(\mathbf{t})$ denote the maximal height of a vertex, and let $\mathbb{P}^{(p)}$ denote its probability law. Uniform height-tail estimate. As $\lambda\to\infty$,

$$\sup_{p\geq 1}\mathbb{P}^{(p)}\left(\mathsf{Height}(\mathbf{t})\geq\lambda\sqrt{p}\right)=O(\lambda^{-6}).$$

This estimate is proposed as an approach to proving that the components removed by the cut-off have negligible height, thereby ruling out long and thin tentacles in the metric approximation of Boltzmann triangulations. The source presents it as an estimate that would imply the desired convergence, without stating that it is known or proved.