On properness of moduli stacks of $D^{\times}$-shtukas over ramified legs

Given a maximal order $\mathcal{D}$ of a central division algebra $D$ over a global function field $F$, we prove an explicit sufficient condition for moduli stacks of $\mathcal{D}^\times$-shtukas to be proper over a finite field (modulo a suitable central action) in terms of the \emph{local invarian