The spheres-tubes-slabs conjecture for the relative isoperimetric problem in the cube

Let $[0,1]^n$ be the unit cube in $\mathbb{R}^n$ with the Euclidean metric. For a measurable set $U\subset [0,1]^n$ of locally finite perimeter, let $\operatorname{Vol}(U)$ denote its $n$-dimensional Lebesgue measure and let $\mathrm{Rel}\;\mathrm{Per}(U)$ denote the $(n-1)$-dimensional Hausdorff measure of the essential boundary of $U$ away from $\partial[0,1]^n$. For $V\in[0,\frac12]$, consider

$$I(V)=\inf\{\mathrm{Rel}\;\mathrm{Per}(U):\operatorname{Vol}(U)=V\}.$$

Here $B^m(r)$ is the closed ball of radius $r$ in $\mathbb{R}^m$ centered at the origin. The spheres-tubes-slabs conjecture. The minimizers for this relative isoperimetric problem, up to isometries of $[0,1]^n$ and sets of measure $0$, are of the form

$$(B^m(r)\cap[0,1]^m)\times[0,1]^{n-m}$$

for some $r\geq0$ and $m\in\{1,\dots,n\}$. This conjecture describes the expected minimizers in the cube: spherical pieces, cylindrical tubes, and slabs. The case $V=\frac12$ is known in all dimensions, but the full conjecture remains open in dimensions $\geq3$.