The non-zero asymptotic excess conjecture for interval exchanges from partial isometries

Let $\Psi$ be an orientation-preserving system of partial isometries, let $\Sigma_\Psi$ be its suspension complex, and let $\widehat\Sigma_{\Psi,\mathbf y}$ be the associated double suspension for admissible parameters $\mathbf y$. Let $T_{\Psi,\mathbf y}$ denote the corresponding interval exchange transformation.

Non-zero asymptotic excess conjecture. If $\Sigma_\Psi$ has non-zero asymptotic excess, then, for every admissible $\mathbf y$, the transformation $T_{\Psi,\mathbf y}$ either is non-minimal or does not fill $\widehat\Sigma_{\Psi,\mathbf y}$.

The conjecture concerns interval exchanges arising from systems of partial isometries and their double suspension surface constructions. The preceding discussion notes that rational independence of the parameters forces non-zero asymptotic excess and suggests that this should obstruct simultaneous minimality and filling; no resolution is given here.