The algebraicity–multiplier group conjecture for quasiperiodic flows

Let $\psi$ be a quasiperiodic flow on an $n$-torus, let $F$ be a real algebraic number field, and let $M_\psi$ denote its multiplier group. The flow is $F$-algebraic when its frequency data, after multiplication by a nonzero real scalar, form a ${\mathbb Q}$-basis of $F$. Let ${\mathfrak o}_F$ be the ring of integers of $F$, and ${\mathfrak o}_F^*$ its group of units.

Algebraicity–multiplier group conjecture. A quasiperiodic flow $\psi$ is $F$-algebraic if and only if $M_\psi$ is a finite index subgroup of ${\mathfrak o}_F^*$.

Examples suggest that $F$-algebraicity and finite index of the multiplier group are closely related, with the conjecture asserting both implications. The supplied text gives examples supporting the claim but does not state that it has been resolved.