Rank-width conjecture for bipartite circle graph pivot-minors

A graph $H$ is a bipartite circle graph if it is both bipartite and a circle graph. Oum's rank-width conjecture. For every bipartite circle graph $H$, there is an integer $r(H)$ such that every graph with no pivot-minor isomorphic to $H$ has rank-width at most $r(H)$.

The conjecture would imply bounded rank-width for graphs excluding an even cycle as a pivot-minor. The analogous assertion fails for odd cycles because every bipartite graph excludes odd cycles as pivot-minors while bipartite graphs can have unbounded rank-width; the supplied text does not resolve the bipartite-circle-graph conjecture.