The ergodic coboundary non-degeneracy conjecture

Let $(\Omega,\mathcal{F},\nu)$ be a probability space, and let $\hat{\sigma},\hat{\tau}:\Omega\to\Omega$ be commuting, measurable, $\nu$-invariant maps. Let $\varphi,\zeta,\delta:\Omega\to\mathbb{R}$ be measurable functions with $\zeta,\delta\geq 0$. Suppose that, $\nu$-almost everywhere,

$$\varphi\circ\hat{\sigma}-\varphi+\zeta\circ\hat{\tau}-\zeta\geq\delta.$$

Ergodic coboundary non-degeneracy conjecture. Then $\delta=0$ $\nu$-almost everywhere.

This is a proposed generalization of an ergodic-theoretic proposition. If true, it would remove a non-degeneracy restriction from the paper's extended-case results and imply its main theorem in that setting; the status of the conjecture remains open.