Minimality criterion for Veech $N$-example flows and interval exchanges

Let $X_N$ be the Veech $N$-example surface, with directional flow $\phi_t^{\theta}$, and let $T_f$ and $T_{1-f}$ be the associated interval exchange transformations. Write the slope parameter as $\alpha$ and the surface parameter as $\beta$. For integers $m,n$, interpret $(N,m,n)\neq 1$ as the condition that $N,m,n$ have a common divisor greater than $1$. Minimality conjecture. If $\alpha$ is irrational, then the flow $\phi_t^{\theta}$ is not minimal on $X_N$ if and only if $\alpha=(n+\beta)/m$; $T_f$ is not minimal if and only if $\beta=m\alpha+n$; and $T_{1-f}$ is not minimal if and only if $\beta=m\alpha+n+1$, for some $m,n\in\mathbb Z$ with $(N,m,n)\neq1$. The conjecture extends the known obstruction criteria for Veech examples; the source states that the “if” directions are known and that some cases are proved later in the paper, while the full equivalences are not established there.