The consecutive-curvature sequence 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. A sequence of spheres is consecutive tangent if each sphere is tangent to the next one.

Consecutive-curvature sequence conjecture. There is a sequence of consecutive tangent spheres

$$(b_0,b_1,\ldots)\subset \Omega(\mathcal{B}_{\mathcal{R}^4})$$

such that, for every $n\in\mathbb N$, the curvature of $b_n$ is $n$.

This is a stronger, structured version of the preceding claim: it requires all natural-number curvatures to occur along one tangent sequence. The source presents it as an open conjecture motivated by numerical experiments.