Multiplier-group distinction conjecture for algebraic and transcendental quasiperiodic flows

Let $\phi$ be a quasiperiodic flow on $T^n$ generated by a constant vector field $X$. Call $\phi$ algebraic if it is $F$-algebraic for a real algebraic number field $F$ of degree $n$ over $\mathbb{Q}$, and call it transcendental otherwise. Write $\rho_\phi(S_\phi)$ for the multiplier group of $\phi$, and let $\mathfrak{o}^*_F$ be the group of units of the ring of integers in $F$. The multiplier-group distinction conjecture. The multiplier group distinguishes the two classes as follows: (a) $\phi$ is algebraic if and only if $\rho_\phi(S_\phi)$ is a finite-index subgroup of $\mathfrak{o}^*_F$ for a real algebraic number field $F$ of degree $n$ over $\mathbb{Q}$; and (b) $\phi$ is transcendental if and only if $\rho_\phi(S_\phi)=\{1,-1\}$. The conjecture is motivated by the contrasting arithmetic examples in the paper, including an algebraic flow with a proper finite-index multiplier subgroup and a transcendental flow with multiplier group $\{1,-1\}$. The supplied text does not state that this distinction conjecture is proved or refuted.