Characterization of conformal factors for flat hexagonal triangulations

Let a hexagonally triangulated flat plane be conformal to the regular triangulation, and let $w$ be its PL conformal factor. The factor $w$ is linear when it is the restriction of a linear function on $\mathbb{C}$ to the vertices of a regular triangulation.

Characterization conjecture. A hexagonally triangulated flat plane is conformal to the regular one if and only if the PL conformal factor $w$ is linear.

This is the equivalent formulation of the preceding two-parameter classification claim. The source does not provide evidence that it has been resolved.