The bipartisan graph quasi-parity conjecture

A bipartisan graph is a graph such that neither it nor its complement has an odd hole, a long prism, a double-diamond, or $L(K_{3,3}\setminus e)$. A quasi-parity graph is a graph in which every induced subgraph or its complement has an even pair.

Maffray–Thomas conjecture. Every bipartisan graph is a quasi-parity graph.

The conjecture was proposed as a possible replacement for the final part of the proof of the strong perfect graph theorem for bipartisan graphs. The supplied status evidence says that this claim is false.