Number of homotopically inequivalent loops that pairwise intersect at most once

Question

This is a follow-up to my previous question, "Number of homotopically inequivalent loops on a surface":

Q. What is the largest number of homotopically inequivalent, simple closed curves, each pair of which intersects in at most one point, that can be drawn on a surface $S$ of genus $g$?

The previous question required the loops to be pairwise disjoint. @LeeMosher answered: $3g - 3$. For $g=4$, allowing the loops to intersect increases the count from $3g{-}3=9$ to at least $13$:

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          ^{Figure by Jeff Erickson here.}

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But I have no confidence that $13$ is the max here. I would be interested to know if there is some general theory that bounds the loops according to their intersections.