Asymptotic linearity conjecture for the extremal edge function $e_\ell(n)$

Fix an integer $\ell\geq 0$. Let $e_\ell(n)$ denote the maximum number of edges in a subgraph of $D_n$ with local crossing number $\ell$, and let $C_\ell$ be the coefficient defined by the paper's piecewise formula. Asymptotic linearity conjecture. For any $n\geq 3$,

$$e_\ell(n)=C_\ell\cdot n+O(1).$$

The preceding bounds place $e_\ell(n)$ between a lower bound with coefficient $C_\ell$ and a crossing-lemma upper bound. The conjecture asserts that the lower-bound coefficient is asymptotically exact; the supplied context gives no resolution.