Density bound conjecture for collectively jammed triangular-torus packings

Let $n$ equal disks be packed in a triangular torus. A packing is collectively jammed if, with the lattice and packing radius fixed, its only continuous motions are continuous translations. Let $\delta_{\Delta}$ denote the density of the triangular packing, with

$$\delta_{\Delta}=\frac{\pi}{\sqrt{12}}.$$

Collectively jammed density conjecture. If such a packing is collectively jammed and is not a triangular packing, then its density is at most

$$\delta=\frac{n}{n+1}\delta_{\Delta}=\frac{n}{n+1}\frac{\pi}{\sqrt{12}},$$

and the maximal density is achieved only by the triangular packing with one disk removed.

This conjecture is closely related to the Density Gap Conjecture, but restricts attention to collectively jammed packings. The source presents it as an open conjecture and gives evidence rather than a proof.