Strict inequality between the 13-crossing and 5-crossing numbers

Let $K$ be a knot, and let $c_n(K)$ denote its $n$-crossing number, the minimum number of $n$-fold crossings in a projection of $K$. Strict inequality conjecture. For all knots $K$ such that $c_5(K)\geq 2$, one has

$$c_{13}(K)<c_5(K).$$

This conjecture predicts that, except for knots with $c_5(K)<2$, the 13-crossing number is strictly smaller than the 5-crossing number. Its resolution is not specified in the supplied source material.