Constant-energy minimal spectral partitions for symmetric planar domains

Let $\Omega$ be a disk, a square, or an equilateral triangle, let $k$ be the number of cells, and let $\mathfrak L_{k,p}(\Omega)$ denote the optimal $p$-norm energy of a $k$-partition. Symmetric-domain partition conjecture. The following numerical observations are conjectured: (i) if $\Omega$ is a disk and $k\in\{2,3,4,5\}$, then $\mathfrak L_{k,p}(\Omega)$ is constant with respect to $p$, and an optimal partition consists of $k$ angular sectors of opening $2\pi/k$; (ii) if $\Omega$ is a square and $k\in\{2,4\}$, then $\mathfrak L_{k,p}(\Omega)$ is constant with respect to $p$, with a minimal $2$-partition given by two equal rectangles or two equal right-isosceles triangles, and with the minimal $4$-partition composed of four squares; (iii) if $\Omega$ is an equilateral triangle and $k=n(n+1)/2$ with $n\geq2$, then $\mathfrak L_{k,p}(\Omega)$ is constant with respect to $p$, and a minimal $k$-partition consists of $3$ equal quadrilaterals, $3(n-2)$ pentagons, and $(n-2)(n-3)/2$ regular hexagons. These are numerical observations, and the source relates the third case to the honeycomb conjecture; no proof of the full collection is supplied.