The curvature realization conjecture for the 24-cell Apollonian packing

Let $\mathcal{B}_{\mathcal{R}^4}$ be the polytopal sphere packing obtained by rescaling a $1$-CBP projection of the $24$-cell by a factor of $\frac{1}{\sqrt{3}}$, and let $\Omega(\mathcal{B}_{\mathcal{R}^4})$ denote its associated packing. Write $\mathbb N$ for the set of natural numbers.

Curvature realization conjecture. The set of curvatures of $\Omega(\mathcal{B}_{\mathcal{R}^4})$ is $\mathbb N$.

Numerical experiments in the source suggest that every integer occurs without any modulo restriction. The statement is presented as one of two conjectures and remains open.