Spin double ramification cycle conjecture for strata of differentials

Let $g,n\geq 0$, let $k>0$, and let $a\in\mathbb{Z}^n$ satisfy $|a|=k(2g-2+n)$, $a\notin k\mathbb{Z}_{>0}^n$, and the condition that all entries of $a$ and $k$ are odd. For odd star graphs and twist data, let ${\rm Star}_g(a)^{\rm odd}$ be the set of star graphs with odd twists at all half-edges; write $m(\Gamma,I)$ for the associated multiplicity, $V_{\rm Out}$ for the outer vertices, ${\rm Aut}(\Gamma,I)$ for the automorphism group, and $\zeta_{\Gamma *}[\overline{\mathcal{M}}_{\Gamma,I}]^{\rm spin}$ for the pushed-forward spin boundary cycle. Spin double ramification conjecture. One has

$$\sum_{(\Gamma,I)\in {\rm Star}_g(a)^{\rm odd}}\frac{m(\Gamma,I)}{k^{|V_{\rm Out}|}|{\rm Aut}(\Gamma,I)|}\,\zeta_{\Gamma *}[\overline{\mathcal{M}}_{\Gamma,I}]^{\rm spin}={\rm DR}_g^{\rm spin}(a).$$

This conjectural identity is the spin analogue of the relationship between strata cycles and double ramification cycles. It is intended to supply the remaining relations needed for the spin version of the intersection-theoretic framework; the stated special cases were checked computationally, while the general identity remains open.