The universal deviation inequality for good drawings of complete graphs

Let $D$ be a good drawing of the complete graph $K_n$, let $\mathcal{F}(D)$ denote its set of faces, and fix a face $F\in\mathcal{F}(D)$. Let $\hat{\Delta}_{\lfloor n/2\rfloor-2}(D)$ and $\bar{\Delta}_{\lfloor n/2\rfloor-2}(D)$ denote the corresponding deviation quantities defined for $D$ with respect to $F$.

Universal deviation inequality. For every face $F\in\mathcal{F}(D)$,

$$\hat{\Delta}_{\lfloor n/2\rfloor-2}(D)\geq \bar{\Delta}_{\lfloor n/2\rfloor-2}(D).$$

The paper reports that this condition held for all drawings and faces inspected and proposes it for all good drawings of complete graphs. Its general validity is left open.