Large-deviation conjecture for the typical Poisson zero-cell vertex number

Let $\mathcal Z_d^{\text{typ}}$ denote the typical Poisson zero cell, and let $f_0(\mathcal Z_d^{\text{typ}})$ be its number of vertices. Large-deviation conjecture. As $d\to\infty$, the sequence of random variables

$$\left(\frac{1}{d}\log f_0(\mathcal Z_d^{\text{typ}})\right)_{d\in\mathbb N}$$

satisfies a large deviation principle on $\mathbb R$ with speed $d$ and a certain non-degenerate rate function. This proposed behavior is consistent with the known first and second moments, which rule out convergence after normalization by the mean to a positive random variable together with all moments; the source gives no resolution.