Invariant reciprocal area sum for lateral harmonic polygons

Let $\mathcal{H}$ be a Poncelet family of $N$-gons interscribed between two homothetic, concentric ellipses, and let $A_1$ and $A_2$ be the areas of the two lateral harmonic polygons obtained as polar images with respect to circles centered at the two foci of the inner ellipse. Lateral-area invariant conjecture. For any odd $N$, the quantity

$$\frac{1}{A_1}+\frac{1}{A_2}$$

is invariant. The formula is proved for $N=3$ and $N=5$, while the assertion for every odd $N$ is presented experimentally and remains open.