This is the 5th research thread of polymath3 studying the polynomial Hirsch conjecture. As you may remember, we are mainly interested in an abstract form of the problem about families of sets. (And a related version about families of multisets.)
There are several reasons why the positive direction is more tempting than the negative one. (And as usual, it does not make much of a difference which direction you study. The practices for trying to prove a statement and trying to disprove it are quite similar.) But perhaps we should try to make also some more pointed attempts towards counterexamples?
Over the years, I devoted much effort including a few desperate attempts to try to come up with counterexamples. (For a slightly less abstract version than that of EHRR.) I tried to base one on the Towers of Hanoi game. One can translate the positions of the game into a graph labelled by subsets. But the diameter is exponential! So maybe there is a way to change the “ground set”? I did not find any. I even tried to look at games (in game stores!) where the player is required to move from one position to another to see if this leads to an interesting abstract example. These were, while romantic, very long shots.
Two more things: First, I enjoyed meeting in Lausanne for the first time Freidrich Eisenbrand, Nicolai Hahnle, and Thomas Rothvoss. (EHR of EHRR.) Second, Oliver Friedmann, Thomas Dueholm Hansen, and Uri Zwick proved (mildly) subexponential lower bounds for certain randomized pivot steps for the simplex algorithm. We discussed it in this post. The underlying polytopes in their examples are combinatorial cubes. So this has no direct bearing on our problem. (But it is interesting to see if geometric or abstract examples coming from more general games of the type they consider may be relevant.)
So let me summarize PHC4 excitements and, as usual, if I missed something please add it.