Asymptotic enumeration of minimally intersecting filling curve systems on closed surfaces

Let $S_g$ be the closed oriented surface of genus $g \geq 2$, and let $Γ$ be a finite collection of closed curves on $S_g$ that fills, in the sense that its complement is a union of disks. The total number of double points of such a $Γ$ in minimal position is at least $2g-1$; we call $Γ$ minimally i