Conjecture on conserved half-angle tangents for polar Harmonic families

Let $N>3$, and consider the polar image of the Harmonic family, a Poncelet family of $N$-gons. For each polygon, let $\theta_i$ denote its relevant angles, so that the half-angle tangents are $\tan(\theta_i/2)$. Polar-Harmonic conservation conjecture. The sum of half-angle tangents is conserved for the polar image of the Harmonic family, for all $N>3$. The statement extends the displayed triangle case to the higher-$polygon$ Harmonic family; the source gives no proof or resolution.