Kalai's 3^d conjecture

Another three-dimensional convex polyhedron, the regular octahedron, is also centrally symmetric, and has 6 vertices, 12 edges, 8 triangular sides, and 1 solid: 6 + 12 + 8 + 1 = 27 = 33.

[3][4] Indeed, these two previous papers were cited by Kalai as part of the basis for making his conjecture.

[1] Another special class of polytopes that the conjecture has been proven for are the Hansen polytopes of split graphs, which had been used by Ragnar Freij, Matthias Henze, and Moritz Schmitt et al. (2013) to disprove the stronger conjectures of Kalai.

All three conjectures claim the Hanner polytopes to minimize certain combinatorial or geometric quantities, have been resolved in similar special cases, but are widely open in general.

In particular, the full flag conjecture has been resolved in some special cases using geometric techniques.