Odd Hadwiger's conjecture

Let $(G,-)$ be the signed graph obtained from a graph $G$ by assigning a negative sign to every edge, and let $(K_t,-)$ be the all-negative signed complete graph. A signed graph minor is obtained by vertex deletion, edge deletion, contraction of positive edges, and switching at vertices. Odd Hadwiger's conjecture. If $(G,-)$ is $(K_t,-)$-minor-free, then

$$\chi(G)\leq t-1.$$

This is a strengthening of Hadwiger's conjecture introduced independently by Gerards and Seymour. The cases $t=3$ and $t=4$ are known, the case $t=5$ implies the four-color theorem, and all remaining cases are open according to the source.