Parity conjecture for the loci W_{d^2}[n]

Let $W_{d^2}[n]$ be the locus under consideration, and for odd $n$ let $W_{d^2}^{\epsilon}[n]$, with $\epsilon\in\{0,1\}$, denote its two parity-labelled subloci. Parity conjecture. Provided that $(d,n)\neq(2,1),(3,1),(4,1)$ or $(5,1)$, the locus $W_{d^2}[n]$ is irreducible when $n$ is even and consists of exactly the two irreducible components $W_{d^2}^{\epsilon}[n]$ for $\epsilon\in\{0,1\}$ when $n$ is odd. This conjecture concerns the component structure of arithmetic Teichmüller loci and is used conditionally in the paper; no resolution is stated.