Spin $k$-twisted double ramification conjecture

Let $g,n\geq 0$ and let $k$ be a positive odd integer. Let $\mu=(m_1,\ldots,m_n)\in\mathbb{Z}^n\setminus k\mathbb{Z}^n_{\geq 0}$ satisfy

$$\sum_i a_i=k(2g-2),$$

with all entries of $\mu$ even. Let $\operatorname{DR}^{\operatorname{spin}}_g(a_\mu)$ be the spin $k$-twisted double ramification cycle, let $SG_1^{\operatorname{odd}}(\mu)$ be the subset of $SG_1(\mu)$ consisting of level graphs whose edge enhancements are all odd, and let $\kappa_e$, $N_0$, and $\xi_{\Delta*}$ have the meanings associated with the corresponding enhanced level graph. Spin $k$-twisted double ramification conjecture. One has

$$\operatorname{DR}^{\operatorname{spin}}_g(a_\mu)=\sum_{\Delta\in SG_1^{\operatorname{odd}}(\mu)}\frac{\prod_{e\in E(\Delta)}\kappa_e}{k^{N_0}|\operatorname{Aut}(\Delta)|}\xi_{\Delta*}\left[\big[\overline{\mathcal{H}}_{g(v)}^k(\mu[v],\kappa[v]-k)\big]^{\operatorname{spin}}\cdot\prod_{v\in V(\Delta^\bot)}\big[\overline{\mathcal{H}}_{g(v)}\left(\frac{\mu[v]}{k},\frac{\kappa[v]-k}{k}\right)\big]^{\operatorname{spin}}\right].$$

This is presented as the spin analogue of the proved formula for ordinary $k$-twisted double ramification cycles. The source reports agreement with computed classes only in a specified range, and gives no proof of the spin conjecture.