The integral Chow rings of the moduli stacks of hyperelliptic Prym pairs II

This paper is the second in a series devoted to describing the integral Chow ring of the moduli stacks $\mathcal{RH}_g$ of hyperelliptic Prym pairs. For fixed genus $g$, the stack $\mathcal{RH}_g$ is the disjoint union of $\lfloor (g+1)/2 \rfloor$ components $\mathcal{RH}_g^n$ for $n = 1, \ldots, \lfloor (g+1)/2 \rfloor$. In this paper, we give presentations and compute the integral Chow rings of the components $\mathcal{RH}_g^{(g+1)/2}$ for odd $g$. As an application, we also obtain presentations and Chow rings for all irreducible components of the moduli stack of hyperelliptic Spin curves of odd genus. An intermediate result of independent interest is the computation of the integral Chow ring of the moduli stack of unordered pairs of divisors of the same even degree in $\mathbb{P}^1$.