The problem naturally fits in the context of breakpoint graphs (per Peter Taylor's observation), which makes it possible to obtain a differential equation for the generating function $$H(x,u,s_1,s_2,\dots) := \sum_{\ell\geq 0} x^{2\ell} \sum_{\tau\in S_{2\ell}} u^{\xi^{(\ell)}(\tau)} \prod_{i=1}^{\#(\tau)} s_{|c_i^{(\tau)}|}$$ by modification of the proof of Lemma 3.2 in my paper https://arxiv.org/abs/1503.05285 similarly how it was done for function $G(x,u,s_1,s_2,\dots)$ in Theorem 4.1. In fact, $H$ represents an analog of function $G$, with just a minor complication that it runs only over even $n=2\ell$.

Then it may be possible to obtain an explicit expression for $H$ and/or derive the anticipated identity directly from the differential equation. The approach can potentially be extended to other values of $\kappa$. Please let me know if you are interested to collaborate on working out details in this direction.

The problem naturally fits in the framework of breakpoint graphs (per Peter Taylor's observation), which makes it possible to obtain a differential equation for the generating function $$H(x,u,s_1,s_2,\dots) := \sum_{\ell\geq 0} x^{2\ell} \sum_{\tau\in S_{2\ell}} u^{\xi^{(\ell)}(\tau)} \prod_{i=1}^{\#(\tau)} s_{|c_i^{(\tau)}|}$$ by modification of the proof of Lemma 3.2 in my paper https://arxiv.org/abs/1503.05285 similarly how it was done for function $G(x,u,s_1,s_2,\dots)$ in Theorem 4.1. In fact, $H$ represents an analog of function $G$, with just a minor complication that it runs only over even $n=2\ell$.

Then it may be possible to obtain an explicit expression for $H$ and/or derive the anticipated identity directly from the differential equation. This approach can potentially be extended to other values of $\kappa$. Please let me know if you are interested to collaborate on working out details in this direction.