The asymptotic cardinality conjecture for systems of simple loops

Let $S_g$ be a closed, orientable surface of genus $g$, and let $k$ be a non-negative integer. A $k$-system is a set of simple loops on $S_g$ in which no two are homotopic or intersect in more than $k$ points. For functions depending on $k$, write $f\sim_k g$ when each of $f\lesssim_k g$ and $g\lesssim_k f$ holds, with the implicit constants depending only on $k$.

Asymptotic cardinality conjecture. The maximum cardinality of a $k$-system of simple loops on $S_g$ is

$$\sim_k g^{k+1}.$$

This conjecture predicts that the maximum size of such a system has the same order of growth as the corresponding extremal problem for simple arcs. The paper proves an upper bound of order $g^{k+1}\log g$, which leaves a logarithmic factor between the known upper bound and the conjectured growth rate.