The transcendentality–multiplier group conjecture for quasiperiodic flows

Let $\psi$ be a quasiperiodic flow, and let $M_\psi$ denote its multiplier group. The flow is transcendental when it is not $F$-algebraic for any real algebraic number field $F$.

Transcendentality–multiplier group conjecture. A quasiperiodic flow $\psi$ is transcendental if and only if

$$M_\psi=\{1,-1\}.$$

The examples in the supplied text show the asserted behavior for particular flows and motivate the equivalence, but no resolution is given in the text.