I just came about the Firoozbakht's conjecture, and read that it is believed to be wrong, as it would contradict some heuristic methods. However, the conjecture is numerically verified for $p_n<10^{18}$.
Are there other examples of mathematical conjectures, that are believed to be false, however no counterexample has been found (even there was some effort to do it)?
A lovely conjecture of Geoffrey Shephard states that for any convex 3-dimensional polytope there is a spanning tree T such that if one cuts along the edges of T then one can unfold the surface of the polytope to obtain a plane polygon where none of the faces of the polytope overlap.
I don't know if most people believe this to be false but for no good reason other than that I would like it to be true, I believe it to be true. It is known that if one takes a "random" polytope that in a precise sense that random unfoldings do overlap. (See C. Schevon and J. O'Rourke. A conjecture on random unfoldings. Technical Report JHU-87/20, Johns Hopkins Univ., Baltimore, MD, July 1987.) nearly always as the number of vertices of the polyhedron increases.
On the other hand a student of Ziegler (Schlickenrieder, Wolfram. "Nets of polyhedra." Master's Thesis, Technische Universität Berlin (1997) showed that for a large number of choices of picking an "appealing" spanning tree to cut along in the hope of getting a non-overlapping unfolding, one could find a polytope where this appealing choice of tree did not work. So, it appears that for each polytope one needs to find "a needle in a haystack" tree T to get an unfolding that avoids overlaps.
If this conjecture were resolved positively it would appear that it would require some new insight into the nature of convex polytopes.
