The crossing-number bound for 14-stick knots in the simple hexagonal lattice

Let $[K]$ be a knot type, let $s_{sh}[K]$ denote its stick number in the simple hexagonal lattice, and let $c[K]$ denote its crossing number. Crossing-number bound conjecture. For a knot type $[K]$ such that $s_{sh}[K] \leq 14$, then $c[K] \leq 5$. This would bound the crossing number of every knot type admitting a simple-hexagonal-lattice presentation with at most 14 sticks; the source says that approaching the claim likely requires improved lower bounds for 6-crossing knots in the cubic lattice, and gives no resolution.