MucMullen's $g$-conjecture aims the classify possible $f$-vectors of simplicial $d$-spheres. The $g$-conjecture has been proven for polytopal spheres and for simplicial spheres of dimension $d < 5$. I am wondering what computer searches have been done to verify (if you are an optimist) or search for counter examples (if you are a pessimist) of the $g$-conjecture.

The only attempt I know of was by Björner and Lutz and was reported in the paper Simplicial manifolds, bistellar flips and a 16-vertex triangulation of the Poincaré homology 3-sphere from 2000. Here it appears Björner and Lutz looked at some (non-PL) simplicial spheres of dimension $d = 5,6$.

I am curious if there have been any more attempts since 2000, and (perhaps too ambitiously) if anyone has been able to make a statement like:

The $g$-conjecture holds for all simplicial $d$-spheres with at most $n$ vertices.

for some nontrivial $d$ and $n$. Of course one could replace "simplicial $d$-shpere" with some class between simplicial spheres and polytopal spheres and that would also be interesting.