The 6-color conjecture for signed triangular grids

A signed graph is a graph whose edges are assigned positive or negative signs; its chromatic number is the least order of a signed target graph admitting a signed graph homomorphism. A signed triangular grid is a finite induced subgraph of the graph associated with the tiling of the plane by equilateral triangles.

Signed triangular-grid chromatic-number conjecture. The chromatic number of every signed triangular grid is $6$.

The paper gives a construction showing that signed triangular grids in which every $C_4$ is unbalanced can be colored with $6$ colors, making this extremal case particularly relevant. The conjecture concerns the exact chromatic number of all signed triangular grids and is not resolved in the supplied text.