The asymptotic hexagonal limit conjecture for spectral equipartitions

Let $\Omega$ be a bounded domain in $\mathbb{R}^2$, and let $\mathfrak L_k(\Omega)$ denote the infimum of the maximal first Dirichlet eigenvalue over all $k$-partitions of $\Omega$. Write $A(\Omega)$ for the area of $\Omega$, and let $\mathrm{Hexa}_1$ be the regular hexagon in $\mathbb{R}^2$ with area $1$. The asymptotic hexagonal limit conjecture. The limit of $\mathfrak L_k(\Omega)/k$ as $k\to+\infty$ exists, and

$$A(\Omega)\lim_{k\rightarrow +\infty} \frac{\mathfrak L_k(\Omega)}{k}=\lambda(\mathrm{Hexa}_1)\;.$$

The conjecture identifies the universal asymptotic energy density with that of the unit-area regular hexagon; the preceding asymptotic upper bound is consistent with this claim, but the existence and exact value of the limit are not established here.