Regular truncated-octahedron conjecture for unit-volume parallelohedra

Let $P$ be a three-dimensional parallelohedron of unit volume, and let $\operatorname{surf}(P)$ denote its surface area. Regular truncated-octahedron conjecture. Among three-dimensional, unit-volume parallelohedra $P$, $\operatorname{surf}(P)$ is minimal if and only if $P$ is a regular truncated octahedron. This is a parallelohedral version of the equal-volume, minimum-surface-area problem for space mosaics; the source notes that no subfamily of mosaics solving Kelvin's problem was known there.