The Hausdorff-dimension-zero conjecture for exceptional bigeodesic times

Fix a dynamical last passage percolation model, and let $\mathscr{T}$ be the set of times at which non-trivial bigeodesics exist. Hausdorff-dimension-zero conjecture. The set $\mathscr{T}$ almost surely has Hausdorff dimension $0$. The conjecture expresses the expectation that, even if bigeodesics occur at exceptional dynamical times, those times are as sparse as possible in terms of Hausdorff dimension; the paper's main theorem gives a subpolynomial lower bound for the probability of times admitting unusually long geodesics but does not resolve this conjecture.