For some early very related classifications see
"The classification of simplicial 3-spheres with nine vertices into polytopes and nonpolytopes" By Altshuler, Bokowski, Steinberg
Neighborly 4-polytopes and neighborly combinatorial 3-manifolds with ten vertices, By Altshuler, Canad J Math 1977
Neighborly 4-polytopes with 9 vertices A Altshuler, L Steinberg - Journal of Combinatorial Theory, Series A, 1973
Enumeration of 3-manifolds with 9 vertices Altshuler and Steinberg
Neighborly 2-manifolds with 12 vertices A Altshuler, J Bokowski, P Schuchert - journal of combinatorial theory, …, 1996
The complete enumeration of the 4-polytopes and 3-spheres with eight vertices
A Altshuler, L Steinberg - Pacific journal of mathematics, 1985