A conjecture on convex polyhedra

We select a class of pyramids of a particular shape and propose a conjecture that precisely these pyramids are of greatest surface area among the closed convex polyhedra having evenly many vertices and the unit geodesic diameter. We describe the geometry of these pyramids. The confirmation of our conjecture will solve the “doubly covered disk” problem of Alexandrov. Through a connection with Reuleaux polygons we prove that on the plane the convex $n$-gon of unit diameter, for odd $n$, has greatest area when it is regular, whereas this is not so for even $n$.