The odd signable graph conjecture for induced $K_{3,3}$ minors

An odd signable graph is a graph with no theta, prism, or even wheel as an induced subgraph; an even-hole-free graph is a graph with no induced cycle of even length. An induced minor is obtained by deleting vertices and edges and contracting connected vertex sets.

Odd signable graph conjecture. If $G$ is an odd signable graph (in particular, if $G$ is an even-hole-free graph), then $G$ does not contain $K_{3,3}$ as an induced minor.

The conjecture is motivated by the structural similarities between odd signable graphs and even-hole-free graphs. It would, in particular, show that even-hole-free layered wheels do not contain $K_{3,3}$ as an induced minor; this is not known in the source.