The odd-square count conjecture for primitive square-tiled surfaces in $\mathcal{H}(2)$

Let $n$ be an odd integer with $n\geqslant 5$. Let $a_n$ and $b_n$ denote the cardinalities of the two discrete orbits of primitive $n$-square-tiled surfaces in ${\mathcal{H}}(2)$, equivalently the indices in $\operatorname{SL}(2,\mathbf Z)$ of their stabilisers. As usual, $p$ ranges over the prime divisors of $n$. The odd-square count conjecture. The orbit cardinalities are

$$a_n = \frac{3}{16}(n-1)n^2\prod_{p\mid n}\left(1-\frac{1}{p^2}\right),\qquad b_n = \frac{3}{16}(n-3)n^2\prod_{p\mid n}\left(1-\frac{1}{p^2}\right).$$

This gives the individual sizes of the two orbits whose total is the known count of primitive $n$-square-tiled surfaces in ${\mathcal{H}}(2)$. The source presents the formula as a conjecture, and no resolution is supplied in the provided text.