Divisibility conjecture for hyperbolic alternating knots with the strong roller-coaster property

Let $K$ be a hyperbolic, alternating knot. Say that $K$ has the strong roller-coaster property when it satisfies the property defined in the paper, and let the crossing number of $K$ be the number of crossings in a minimal diagram.

Divisibility conjecture. If $K$ has the strong roller-coaster property, then the crossing number of $K$ is divisible by four.

The theorem preceding this conjecture verifies it for alternating, hyperbolic knots with at most 12 crossings. The general statement remains open.