Hexagonal conjecture for asymptotic minimal partition energies

Let $\Omega$ be a bounded planar domain, and let $\mathfrak L_k(\Omega)$ be the energy of a minimal $k$-partition of $\Omega$. Let $\hexagon$ be the regular hexagon of area $1$, and let $\lambda(\hexagon)$ be its ground-state Dirichlet eigenvalue. Hexagonal conjecture. The limit exists and satisfies

$$|\Omega|\lim_{k\to+\infty}\frac{\mathfrak L_k(\Omega)}{k}=\lambda(\hexagon).$$

The conjecture expresses the expected hexagonal asymptotic shape of optimal partitions and remains open in this generality.