Longest minimal length partition conjecture

Let $d\geq 2$, let $v_d$ denote the volume of the unit ball in $\mathbb{R}^d$, and let $\Omega\subset\mathbb{R}^d$ be a domain. For $c\in(0,1)$, let $L(\Omega,c)$ be the minimal length of a partition of $\Omega$ into two regions of prescribed volumes $c|\Omega|$ and $(1-c)|\Omega|$. For $n>1$ and ${\mathbf c}=(c_i)_{i=1}^n\in\mathbb{R}^n_+$ with $\sum_{i=1}^n c_i=1$, let $PL(\Omega,{\mathbf c})$ be the minimal perimeter of a partition into $n$ regions with prescribed volume fractions $c_i$.

Longest minimal length partition conjecture. Given $c\in(0,1)$, the set $\Omega$ maximizing $L(\Omega,c)$ under the constraint $|\Omega|=v_d$ is the ball. Given $n>1$ and ${\mathbf c}=(c_i)_{i=1}^n\in\mathbb{R}^n_+$ with $\sum_{i=1}^n c_i=1$, the set $\Omega$ maximizing $PL(\Omega,{\mathbf c})$ under the constraint $|\Omega|=v_d$ is the ball.

The conjecture extends the equal-area results to partitions with prescribed, possibly unequal, volumes. Numerical simulations suggest that the disk in two dimensions and the ball in three dimensions maximize the minimal partition length or perimeter, but the general assertion remains open.