Near-existence of bigeodesics in dynamical exponential last passage percolation

It is believed that, under very general conditions, bi-infinite geodesics (or bigeodesics) do not exist for planar first and last passage percolation (LPP) models. However, if one endows the model with a natural dynamics, thereby gradually perturbing the geometry, then it is plausible that there could exist a non-trivial set $\mathscr{T}$ of exceptional times at which such bigeodesics exist. For dynamical exponential LPP, we show that $\mathscr{T}$ is "very close" to being non-trivial; namely, we obtain an $\Omega( 1/\log n)$ lower bound on the probability that there exists a random time $t\in [0,1]$ at which a non-trivial geodesic of length $n$ passes through the origin at its midpoint; note that if the above probability were $\Omega(1)$, then it would imply the non-triviality of $\mathscr{T}$. We conjecture that, even if $\mathscr{T}\neq \emptyset$, it a.s. has Hausdorff dimension exactly zero.