Conjecture that area-minimizing n-hedral tiles are convex

An $n$-hedral tile is an $n$-faced polyhedron that tiles space by congruent copies. A tile is area-minimizing when it has least surface area among $n$-hedral tiles of the prescribed volume.

Convexity conjecture. For any $n$, the least-area $n$-hedral tile is convex.

The conjecture asks whether minimizing surface area under the tiling constraint forces convexity; it remains open in the source.