The signed projective cube homomorphism conjecture for signed planar graphs

Let $(G,\sigma)$ be a signed planar graph, and let $g_{ij}(G,\sigma)$ denote the corresponding signed girth parameter; let $SPC(k)$ be the signed projective cube of dimension $k$, with $g_{ij}(SPC(k))$ its corresponding parameter. Signed projective cube homomorphism conjecture. If

$$g_{ij}(G,\sigma) \geq g_{ij}(SPC(k)),$$

then there is a signed graph homomorphism

$$(G,\sigma) \to SPC(k).$$

This conjecture combines a signed-graph analogue of the homomorphism formulation of the four-color theorem with the corresponding projective-cube homomorphism conjecture. Its status is not resolved in the supplied text.