Guenin's projective-cube conjecture for $K_5$-minor-free signed bipartite graphs

Let $(G,\Sigma)$ be a signed bipartite graph, with unbalanced girth at least $2k$, that has no $(K_5,E(K_5))$-minor. The signed projective cube $SPC(2k-1)$ is the signed graph on vertex set $\mathbb{Z}_2^{2k-1}$ whose positive edges correspond to differences $e_1,\ldots,e_{2k-1}$ and whose negative edges correspond to the all-1 vector $J$. A homomorphism of signed graphs preserves the balance of closed walks. Guenin's $K_5$-minor-free conjecture. Then

$$(G,\Sigma)\to SPC(2k-1).$$

This is described as a stronger conjectural extension of the planar statement, replacing planarity by exclusion of a signed $(K_5,E(K_5))$-minor. The source does not report a proof or disproof.