Endpoint behavior of optimal p-norm spectral-partition energies

Let $\Omega$ be a domain, let $k$ be a positive integer, let $\mathfrak L_{k,p}(\Omega)$ be the optimal $p$-norm energy of a $k$-partition, and let $p_{\infty}(\Omega,k)$ denote the parameter introduced in the source. Endpoint-energy conjecture. One has

$$p_{\infty}(\Omega,k)\in\{1,\infty\},$$

and either $\mathfrak L_{k,p}(\Omega)$ is constant with respect to $p$ or it is strictly increasing with respect to $p$. Moreover, numerical simulations suggest that $p_{\infty}(\Omega,k)=1$ if one of the following holds: $\Omega$ is a disk and $k\in\{2,3,4,5\}$; $\Omega$ is a square and $k\in\{2,4\}$; or $\Omega$ is an equilateral triangle and $k=n(n+1)/2$ with $n\geq1$. The claim summarizes numerical evidence concerning how the optimal energy changes with $p$; it is not presented with a proof, and the source does not provide a general resolution.