Minimal-energy equality conjecture for three, four and five partitions of the square torus

Let $\mathsf{T}(1,1)$ be the square flat torus. Let $\mathsf{H}_3(1)$ and $\mathsf{H}_4(1)$ be the hexagonal tiling domains appearing in Proposition 1, and let $\mathsf Q$ be the square of side $1/\sqrt{5}$ used as the tiling domain of a $5$-partition. The proposition gives

$$\mathfrak{L}_3(\mathsf{T}(1,1))\leq\lambda_1(\mathsf{H}_3(1)),\qquad \mathfrak{L}_4(\mathsf{T}(1,1))\leq\lambda_1(\mathsf{H}_4(1)),$$

and

$$\mathfrak{L}_5(\mathsf{T}(1,1))\leq\lambda_1(\mathsf Q)=10\pi^2.$$

Minimal-energy equality conjecture. The three inequalities in Proposition 1 are equalities.

If true, these candidates would give the exact minimal energies for $3$-, $4$- and $5$-partitions of the square torus. The statement is presented as a conjecture, with only the displayed upper bounds established in the source.