Average-distance conjecture for touching Jordan curves

Let $\mathcal{F}$ be a $k$-touching family of Jordan curves, meaning that no point of the plane lies on more than $k$ curves. For each pair of intersecting curves $a,b\in\mathcal{F}$, let $d(a,b)$ be the number of curves whose bounded region contains exactly one of $a$ and $b$, and define the average distance in $\mathcal{F}$ as the average of $d(a,b)$ over all intersecting pairs.

Average-distance conjecture. For any $k$-touching family $\mathcal{F}$ of Jordan curves, the average distance in $\mathcal{F}$ is at most

$$\frac{k}{2}.$$

The conjecture concerns possibly nonsimple families of touching Jordan curves, in which two curves may intersect at several points. The supplied source text gives no resolution evidence for this claim; its status is therefore open.