Conjectural minimality criterion for Veech extensions with $m=\bar m n+1$

Let $0<\alpha<1$ be irrational, let $T$ be the rotation by angle $\alpha$, viewed as a two-interval exchange with discontinuity $1-\alpha$, and let $T_f$ be the Veech extension from Definition with $q=1$. Suppose $0<\zeta_1=m\alpha-n<1$, where $m=\bar m n+1$ and $\bar m\geq1$. Minimality conjecture. The transformation $T_f$ is minimal if and only if

$$(N,a_1+(\bar m-1)a_2,ma_2)=1.$$

This is presented as a generalization of case $(vi)$ and is supported in the source by hand computations for small values; the other cases of the general criterion are described as unresolved there.