Neighbourly polytopes with few vertices

A family of neighbourly polytopes in $\mathbb R^{2d}$ with $N=2d+4$ vertices is constructed. All polytopes in the family have a planar Gale diagram of a special type, namely, with exactly $d+3$ black points in convex position. These Gale diagrams are parametrized by $3$-trees (trees with a certain additional structure). For all polytopes in the family, the number of faces of dimension $m$ containing a given vertex $A$ depends only on $d$ and $m$.
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