Tightness of intersections for critical percolation crossings

Let $Q$ be a quad and let $\alpha$ be a smooth curve. For a discrete percolation measure $\mu_\eta$, let $N_\eta=N_\eta(Q,\alpha)$ be the minimal possible number of intersections with $\alpha$ for a crossing of $Q$, with $N_\eta=0$ if $Q$ is not crossed. Tightness of intersections. The random variables $N_\eta$ are tight as $|\eta|\to0$. This is presented as a discrete version that is a priori stronger than the finite intersection property; proving either conjecture would simplify the main theorem and provide a finite gluing procedure for crossings.