The zero-condition conjecture for degenerate pyramids

Let $P'$ be a degenerate pyramid whose four base vertices are vectors $p_i\in\mathbb{R}^2$, with top vertex at $0$, and let $p_0=p_4$. Define

$$F(i):=|p_i|\left(1-\frac{\langle p_i-p_{i+1},p_i\rangle}{|p_i||p_{i+1}-p_i|}-\frac{\langle p_i-p_{i-1},p_i\rangle}{|p_i||p_{i-1}-p_i|}\right).$$

The conditions referred to as those of the Clean Condition are the relations $F(1)=-F(2)=F(3)=-F(4)$.

Zero-condition conjecture. The conditions from the Clean Condition are satisfied only when $F(i)=0$.

This conjecture is presented as an apparently easy unresolved question in the discussion of quadrilateral faces. The supplied text gives no proof or resolution.