Conservation of half-angle tangents for polar Harmonic families

Let $N>3$, and consider the Harmonic family, together with its polar image with respect to the circumcircle. For the associated polygon, let $\theta_i$ denote the relevant angles. The sum of half-angle tangents is

$$\sum_i \tan\frac{\theta_i}{2}.$$

Polar Harmonic-family conjecture. The sum of half-angle tangents is conserved for the polar image of the Harmonic family, for all $N>3$.

The claim extends the conservation law established immediately beforehand for the iso-$X_7$ triangle family to the polar image of the $N>3$ Harmonic family. The source gives no resolution or further evidence of its status.