Centred-hexagonal-number conjecture for the smallest hexagon containing disks

Let $n\in\mathbb{N}$ be a centred hexagonal number, and consider the smallest hexagon containing $n$ disks and the smallest hexagon containing $n-1$ disks. Centred-hexagonal-number conjecture. The smallest hexagon containing $n$ disks is the same as that containing $n-1$ disks. The source proves that when $n$ is a centred hexagonal number, the smallest containing hexagon is regular, and proposes this statement as an analogy of the triangular-number conjecture; the asserted equality remains open.