Here is a web page by Christope Weibel on the improved counterexample.
The IPAM webpage contains now slides of some of the lectures. Here are Santos’s slides. The last section contains some recent results on the “width of 5-prismatoids” A prismatoid is a polytope with two facets containing all the vertices. The width of a prismatoid is the number of steps needed to go between these two facets where in each step we move from a facet to an adjacent one. Santos’s counterexample is based on findng 5-dimensional prismatoid with width larger than 5. It is observed that the width of a 5-prismatoid with n vertices cannot exceed 3n/2 and it is shown (by rather involved constructions) that there are examples where the width is as large as .