Strong sausage conjecture for finite parametric sphere packings

Let $B^d$ be the unit ball in $d$-dimensional Euclidean space, let $\delta_\rho(B^d,n)$ be the optimal parametric density for a finite packing of $n$ balls with parameter $\rho>0$, and let $S_n(B^d)$ be a sausage configuration. Let $\delta(B^d)$ denote the optimal density of an infinite packing, and let $\rho_s(B^d)$ and $\rho_c(B^d)$ be the sausage and critical parameters. Strong sausage conjecture. For every $n\in\mathbb{N}$ and every $\rho>0$,

$$\delta_\rho(B^d,n)=\delta_\rho(B^d,S_n(B^d))\quad\text{or}\quad\delta_\rho(B^d,n)<\delta(B^d).$$

In particular,

$$\rho_c(B^d)=\rho_s(B^d).$$

The claim would say that whenever a finite packing does not have an optimal sausage density, its density is strictly below the infinite-packing density. The source presents this as an open conjecture and notes that equality of the two parameters is generally false for convex bodies in dimensions at least three, making the ball case special.