Flatness conjecture for consecutive random polygon subdivisions

Let $L_n$ be the random polygon sequence generated by the consecutive random subdivision, and suppose that $u$ satisfies

$$\mathbb{P}(\xi\in\{0,1\})=0$$

and that the support of $\xi$ contains at least two distinct points in $(0,1)$. A sequence of polygons converges to a flat figure when

$$\lim_{n\to\infty}\frac{\mathcal{A}(L_n)}{\left(\max_{j=1,\dots,d}\|l_j^{(n)}\|\right)^2}=0,$$

where $\mathcal{A}(L_n)$ is the area of $L_n$ and $l_j^{(n)}$ are its side vectors. Flatness conjecture. Under these assumptions, the sequence of polygons $L_n$ converges to a flat figure almost surely as $n\to\infty$. The paper presents this as the main phenomenon it aims to establish partially; the supplied text gives no resolution status beyond the assertion itself.