Fredholmness and finite-dimensional periodic cohomology conjecture

Let $E_{b4}(Z)$ be the operator associated with the elliptic complex on the periodic end, and let $\widehat{H}^*_z(V)$ denote the corresponding periodic cohomology group, with parameters $\delta \in \mathbb{R}$ and $z \in \mathbb{C}$. Fredholmness conjecture. $E_{\delta}(Z)$ is Fredholm with respect to some $\delta \in \mathbb{R}$ if and only if $\widehat{H}^*_z(V)$ is finite dimensional with respect to some $z \in \mathbb{C}$. The conjecture proposes an equivalence between the analytic Fredholm condition and finite-dimensionality of the homological object arising from the filtered periodic complex; the supplied text gives no resolution.