Invariance of cosine sums for bicentric pedal polygons

A bicentric pedal polygon is obtained from a bicentric polygon by taking its pedal polygon with respect to a limiting point, denoted by $\ell_1$ or $\ell_2$. Cosine-sum invariance conjecture. The sum of the cosines of bicentric pedal polygons with respect to either limiting point is invariant, except for the $\ell_1$-pedal in the $N=4$ case. This claim is supported by experimental evidence and is stated as not yet proved; the exceptional $\ell_1$-pedal case for $N=4$ is excluded.