Invariant sine-angle ratio for harmonic polygons

Let $P$ be a harmonic polygon with area $A$, sidelengths $s_i$, and angles $\theta_i$ as used in the paper. Sine-angle invariant conjecture. The quantity

$$\frac{\sum\sin(2\theta_i)}{A}$$

is invariant. Equivalently,

$$\frac{\sum\sin(2\theta_i)}{\sum s_i^2}$$

is invariant. The source gives no resolution or proof for this assertion, so its status remains open.