Cube tilings and alternating links

Consider a planar graph G, and form the lattice of integer-valued flows on G. Is it the case that this lattice embeds into the lattice of integer points in Euclidean space in such a way that each unit cube with integer vertices contains a point of the embedded sublattice?

Consider instead an alternating link L, and form the double-cover of the three-sphere branched along L. Is it the case that this space bounds a smooth four-manifold with trivial rational homology groups?

Under the correspondence that takes L to its Tait graph G, we conjecture that the answers to these two questions are the same. I will explain why a positive answer to the second implies a positive answer to the first using Floer homology.  I will then explain why a positive answer to the first implies a positive answer to the second under the added hypothesis that each unit cube contains a *unique* point of the embedded sublattice.