Rigidity conjecture for locally finite hexagonal square tilings

Let ${S_i\mid i\in I}$ be a locally finite square tiling of the complex plane $\mathbb{C}$ such that each square intersects exactly six others.

Hexagonal square-tiling rigidity conjecture. All squares $S_i$ have the same size.

The conjecture concerns the rigidity of the third regular planar pattern discussed in the paper, namely the hexagonal square tiling. It is proposed at the end of the paper, and no resolution is supplied there.