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.