Necessity of the spectral-gap assumption for approximate measurable bipartiteness

Determine whether the spectral-gap assumption is necessary for an ergodic bounded-degree Borel pmp graph whose adjacency operator has $-M(T_G)$ in its spectrum to be approximately measurably bipartite.

Background

The paper proves that if an ergodic bounded-degree Borel pmp graph has an adjacency operator with a spectral gap and if the negative of its maximum spectral value belongs to the spectrum, then the graph is approximately measurably bipartite. The spectral gap enables the argument to control approximate eigenvectors associated with the negative extremal spectral value and relate their absolute values to a positive extremal eigenvector.

Although the paper notes that ergodicity is necessary, it does not establish whether the spectral-gap hypothesis is merely a technical device or an essential condition. Removing or weakening that hypothesis would extend the spectral characterization of approximate measurable bipartiteness to a broader class of non-regular graphs.