The edge-addition conjecture for rooted phylogenetic networks

Let $N$ be a rooted phylogenetic network on leaf set $X$. The quantities $\ell(N)$ and $e(N)$ measure, respectively, the tree-based proximity of $N$ and the minimum number of directed edges that must be added to make $N$ tree-based. Edge-addition conjecture. There exists a set of directed edges that can be added to $N$ to turn it into a tree-based rooted phylogenetic network on leaf set $X$. Furthermore,

$$\ell(N)\le e(N)\le 2\ell(N).$$

The conjecture asserts both that every rooted phylogenetic network can be made tree-based by adding directed edges and that the required number of additions is at most twice its tree-based proximity. The preceding example shows that the unrooted equality $e(N)=\ell(N)$ does not extend to rooted networks, since there $e(N)=2>1=\ell(N)$.