Isoperimetric conjecture on the three-dimensional cube

Let $\beta\in(0,1]$ and let

$$\mathbf{Q}^3(\beta)=([0,\beta]\times[0,1]^2,\lvert\cdot\rvert^2,\frac{1}{\beta}\mathfrak{m}\llcorner_{[0,\beta]\times[0,1]^2})$$

be the three-dimensional rectangular cube with its uniform probability measure. Write $\bar v$ for relative volume, and let $r$ be chosen so that each listed set has relative volume $\bar v$.

Isoperimetric conjecture on the three-dimensional cube. For every $\bar v\in(0,1)$, at least one of the following sets or its complement is an isoperimetric minimizer in $\mathbf{Q}^3(\beta)$ of relative volume $\bar v$: an eighth ball around a vertex, a quarter cylinder around the short edge $[0,\beta]$, or a half-plane $\{x\in\mathbf{Q}^3(\beta):x_3\leq\bar v\}$.

This is widely believed and has been described as one of the nicest open problems in classical geometry. The paper notes that only relative volumes near $0$, $1/2$, and $1$ were previously established by compactness arguments, while its results prove the conjecture in additional ranges, including the standard cube for $\bar v\leq0.120582$.