Eventual crossing-free-edge conjecture for extremal subgraphs of $D_n$

Let $D_n$ be the graph under consideration, let $e_\ell(n)$ denote the maximum number of edges in a subgraph of $D_n$ with local crossing number $\ell$, and call an edge crossing free if it is crossed zero times. Eventual crossing-free-edge conjecture. For each integer $\ell\geq 0$ there is a positive integer $N_\ell$ such that for all integers $n\geq N_\ell$ there is a subgraph $G$ of $D_n$ with local crossing number $\ell$ such that $G$ has a crossing free edge and $e(G)=e_\ell(n)$. The claim would provide optimal constructions that are parallel compositions of smaller graphs for fixed $\ell$ and sufficiently large $n$; the paper gives no resolution in the supplied context.