Delecroix–Lelièvre's orbit-count conjecture in \mathcal{H}$4$

Let $n$ be an integer greater than $8$, and consider primitive $n$-squared origamis in $\mathcal{H}(4)$, with $\mathcal{H}^{\mathrm{odd}}(4)$ denoting the odd component and $\mathcal{H}^{\mathrm{hyp}}(4)$ the hyperelliptic component. Delecroix–Lelièvre's orbit-count conjecture. The number of $\operatorname{SL}(2,\mathbb{Z})$-orbits is as follows: there are precisely two orbits in $\mathcal{H}^{\mathrm{odd}}(4)$ outside the Prym locus, distinguished by monodromy groups $\operatorname{Alt}(n)$ and $\operatorname{Sym}(n)$; for odd $n$, there are precisely four orbits in $\mathcal{H}^{\mathrm{hyp}}(4)$, distinguished by HLK-invariants $(4,[1,1,1])$, $(2,[3,1,1])$, $(0,[5,1,1])$, and $(0,[3,3,1])$; and for even $n$, there are precisely three orbits in $\mathcal{H}^{\mathrm{hyp}}(4)$, distinguished by HLK-invariants $(3,[2,2,0])$, $(1,[4,2,0])$, and $(1,[2,2,2])$. The source attributes this conjecture to Delecroix and Lelièvre, notes substantial computational evidence, and additionally records the implicit expected orbit growth $\Omega(n^5)$; the classification itself remains conjectural in the supplied text.