The Nase conjecture on triple cumulative edge bounds in simple graph drawings

Let $D$ be a simple drawing of a graph, and let $E_{{\le}{\le}{\le}k}(D)$ denote the number of its $\le\!\le\!\le k$-edges, defined by

$$E_{{\le}{\le}{\le}k}(D)=\sum_{j=0}^{k}E_{{\le}{\le}j}(D)=\sum_{i=0}^{k}{k+2-i\choose 2}E_i(D).$$

Here $E_i(D)$ denotes the number of $i$-edges, and the cumulative notation is obtained by summing the corresponding lower-order edge counts. The Nase conjecture. If $n\ge 3$ and $D$ is a simple drawing of $K_n$, then for every $k$ satisfying $0\le k<n/2-1$,

$$E_{{\le}{\le}{\le}k}(D)\ge 3{k+4\choose 4}.$$

The conjecture is stronger than Hill's conjecture. It is implied by the paper's theorem for simple $x$-monotone drawings, and all examples considered by the authors satisfy it; its validity for arbitrary simple drawings remains open.