On the number of facets of a 2-neighborly polytope

A $d$-polytope $P$ is $2$-neighborly if each $2$ vertices of $P$ determine an edge. It is conjectured that the number $f_0(P)$ of vertices for such polytope does not exceed the number $f_{d-1}(P)$ of facets. The conjecture is separately proved for $d<7$ and for $f_0(P)<d+6$.