Veech-group commensurator conjecture for isogenous interval exchange systems

Let two interval exchange systems be isogenous, and suppose one is minimal and self-induced. Then the other is also minimal and self-induced. Associate to such systems their translation surfaces with pseudo-Anosov automorphisms. Veech-group commensurator conjecture. One associated translation surface is Veech if and only if the other is Veech; in that case, their Veech groups have isomorphic commensurators in $\operatorname{PGL}_2(\mathbb{R})$. This conjecture concerns invariance of the Veech property and the commensurator structure under isogeny; the source gives no resolution.