The cubical minimizers conjecture for the relative isoperimetric problem

Let $[0,1]^n$ be the unit cube in $\mathbb{R}^n$, and let a cubical subset be a compact subset of $[0,1]^n$ whose boundary is contained in finitely many hyperplanes with outer normal vectors in $\{\pm e_1,\dots,\pm e_n\}$, where $e_1,\dots,e_n$ is the standard basis. For $V\in[0,\frac12]$, consider the relative isoperimetric problem of minimizing the perimeter inside the cube among cubical subsets of volume $V$. The cubical minimizers conjecture. Up to isometries of $[0,1]^n$ and sets of measure $0$, the minimizers are of the form

$$[0,a]^m\times[0,1]^{n-m}$$

for some $a\in[0,\frac12]$ and $m\in\{1,\dots,n\}$. This is the cubical analogue of the spheres-tubes-slabs conjecture. The conjecture is presented as an open problem; the supplied text does not give a resolution status.