The generalized Nase conjecture on triple cumulative edge bounds

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, with

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

Here $E_i(D)$ denotes the number of $i$-edges in the drawing. The generalized Nase conjecture. If $k\ge 0$ and $D$ is a simple drawing of a graph with at least ${2k+3\choose 2}$ edges, then

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

This extends the complete-graph conjecture to arbitrary graphs. The authors report no counterexample, but provide no proof in general, so the statement remains open.