Conjecture on weakening the intensity-growth condition for limit theorems

Let $t$ denote the intensity parameter and let $d$ tend to infinity. The Gaussian and Poisson limit theorems above require growth conditions on $t$ because of dimension-dependent factors in their probability-metric bounds. Intensity-growth conjecture. These conditions can be weakened, or possibly dropped completely, by improving the dimension-dependent factors through calculating exactly the integrals arising in the derivative bound theorem, as in the proof of the variance limit lemma. This would enlarge the regimes in which the Gaussian and Poisson approximations hold for the face-counting functional of the high-dimensional random Vietoris–Rips complex.