Can the lattice stick number of a knot be bounded by the stick number of the knot?
The stick number $S(K)$ of a knot $K$ is the fewest number of segments needed to realize it by a simple 3D polygon. The lattice stick number $S_L(K)$ is the fewest segments in a realization in the cubic lattice, with all segments parallel to a coordinate axis. For example, the stick number of the trefoil knot $K=3_1$ is 6, and its lattice stick number is 12 (the latter a result of Huh and Oh from 2005).
My question is whether it is possible to bound $S_L(K)$ by $m S(K)$, where $m$ is some multiplier factor. Ideally $m$ would be a constant, but perhaps it is more realistic to expect it to depend on the complexity of the knot (e.g., on its crossing number $cr(K)$). What I have in mind is replacing each stick in a stick realization by a bounded lattice path.
Addendum. Tracy Hall's clever example below indicates that it is unlikely that $m$ could be a constant.