"Kontsevich conjectured that the number f(G,q) of zeros over the finite field with q elements of a certain polynomial connected with the spanning trees of a graph G is polynomial function of q." (Richard P. Stanley, "Spanning trees and a conjecture of Kontsevich").
It was disproved by Prakash Belkale, Patrick Brosnan "Matroids, motives and conjecture of Kontsevich".
On the other hand the story seems to be open-ended: still there are many interesting cases where the conjecture is true (e.g. "Do Tutte Polynomials Satisfy The Kontsevich Conjecture?", J.Stembridge checked it up graphs with 12 nodes "Counting points on varieties over finite fields related to a conjecture of Kontsevich"). R. Stanley related the conjecture to some natural question of counting symmetric invertible matrices with zeros at some positions (see e.g. S.Sam slides). In early 2000-ies expert Feynman diagrams Eduard Lerner told me that from physicist's point of view original form the conjecture lacks signs. He put things in print much later e.g. here, here, etc.
There seems to be no original text by M.Kontsevich on these things and conjecture is known through the Stanley's paper. It mentions that motivation was related to some ideas related to Feynman diagrams, but there seems to be no much details about that. In general there are some other examples of "polynomial count" varieties (see e.g. refs here MO, here ), but it seems not so clear the natural generality of these examples. So it would be interesting:
Question 1: Can one shed light on motivation of the conjecture ?
Question 2: What modifications are possible to get some other polynomial count varieties (or at least some conjectures on that) ?
