Rational independence of the eigenvalues for odd polygons

Let $n$ be odd, write $n=2m+1$, and let $bbda_0=0,bbda_1,bbda_{n-1}$ be the eigenvalues of the circulant matrix governing the linearized system, with $bbda_j=-bbda_{n-j}$ and, for $j=1,\dots,m$,

$$bbda_j=\sqrt{-1}\cos\frac{\pi}{n}\tan\frac{\pi j}{n}.$$

Rational-independence conjecture. For odd $n\ge 7$, the eigenvalues $bbda_j$, $j=1,\ldots,m$, are rationally independent. This conjecture is used to deduce infinitesimal rigidity of the regular odd polygons; the supplied text gives no resolution of the conjecture.