Finite intersection property for critical percolation crossings

Let $Q$ be a quad and let $\alpha$ be a smooth curve. A crossing of $Q$ is a percolation crossing between the designated boundary arcs of the quad. Finite intersection property. In the scaling limit, conditional on the existence of a crossing, there is almost surely a crossing intersecting $\alpha$ only a finite number of times. The preceding discussion gives positive conditional probability for finitely many intersections, while whether that probability is one is posed as an open conjecture; it would simplify the proof and enable a constructive gluing procedure for crossings.