Bezdek's Rhombic Dodecahedral Conjecture for parallelohedra

Let $P$ be a parallelohedron in $\mathbb R^3$ with unit inradius. Bezdek's Rhombic Dodecahedral Conjecture. The surface area of $P$ is at least

$$12\sqrt{2}\approx 16.97,$$

which is the surface area of the regular rhombic dodecahedron with unit inradius. The source identifies this as the lattice variant of the Strong Dodecahedral Conjecture, which was proved by Hales, while presenting the Rhombic Dodecahedral Conjecture as an existing conjecture.