The parity conjecture for elliptic-cover loci in genus two

Let $d>1$ and $n\geq 1$ be integers. Let $W_{d^2}[n]\subset\mathcal M_2$ be the locus of genus-two Riemann surfaces admitting a primitive degree-$d$ elliptic cover with two critical points whose images differ by a point of order $n$ in the Jacobian of the elliptic curve. The loci $W_{2^2}[1]$ and $W_{3^2}[1]$ are empty, while $W_{4^2}[1]$ and $W_{5^2}[1]$ are irreducible. Parity conjecture. Provided $(d,n)\ne(2,1),(3,1),(4,1),(5,1)$, $W_{d^2}[n]$ is irreducible when $n$ is even, and consists of two irreducible components when $n$ is odd. This conjecture concerns the classification of imprimitive Teichmüller curves in the genus-two moduli space; it is known that every such curve is a component of some $W_{d^2}[n]$, and the conjecture has been proved for $d=2$ and arbitrary $n$, while the general case remains open.