Hexagonal conjecture for the boundary length of minimal spectral equipartitions

Let

$$be a bounded open domain in$$

, let

be a sequence of minimal regular spectral $k$-equipartitions of

, and let

$$denote the length of the boundary set of$$

. Let

be the regular hexagon of area $1$, and let

denote its boundary length. Then

$$\ell(\mathrm{Hexa}_1)=2\sqrt{2\sqrt{3}}=2(12)^{\frac{1}{4}}.$$

Hexagonal boundary-length conjecture. The normalized boundary length satisfies

$$\lim_{k\rightarrow +\infty}\frac{P(\mathcal D_k)}{\sqrt{k}}=\frac{1}{2}\ell(\mathrm{Hexa}_1)\sqrt{A(\Omega)}.$$

This is motivated by the preceding hexagonal conjecture for the asymptotic energy and expresses the expected hexagonal structure of large minimal partitions. The source does not state a resolution.