The 13-stick classification conjecture for 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 $3_1$, $4_1$, $5_1$, and $5_2$ denote the indicated knot types. 13-stick classification conjecture. The only non-trivial knot types $[K]$ with stick number $s_{sh}[K] \leq 13$ in the cubic lattice are $3_1$ and $4_1$. In particular, $s_{sh}(5_1) = s_{sh}(5_2) = 14$. This conjecture would restrict the knot types realizable with at most 13 sticks and determine the stick numbers of the two 5-crossing knots; the supplied text gives no resolution.