The stable-map decomposition formula for powers of the cotangent class

Let $G,d,m$ be as in the moduli problem, let $x$ be the marked point, and let $\overline{\mathcal M}^0_{G,1,d}(\mathbb{CP}^1)$ denote the relevant moduli space of stable maps. For integers $g,n,p,K$ and positive integers $k_1,\dots,k_n$ satisfying the displayed constraints, set

$$G'=G-g-n+1,\qquad \kappa=(k_1,\dots,k_n,1^{d-K}).$$

Let ${\rm DR}_g(\framebox{K-p},-k_1,\dots,-k_n;\widetilde 1^p)$ denote the indicated pushed-forward double ramification class, and let the second factor be the corresponding disconnected relative stable-map virtual class. Stable-map decomposition conjecture.

$$m!\,\psi_x^m\,[\overline{\mathcal M}^0_{G,1,d}(\mathbb{CP}^1)]^{\rm virt}= \sum_{n\geq1}\frac1{n!}\sum_{g=0}^{G-n+1}\sum_{p=0}^g\sum_{K=p}^{m+p}\frac{m!}{p!\,(K-p)!}\,\psi_x^{m-K+p}\sum_{\substack{k_1,\dots,k_n\sum k_i=K}}\prod_{i=1}^n k_i\times {\rm DR}_g(\framebox{K-p},-k_1,\dots,-k_n;\widetilde 1^p)\boxtimes[\overline{\mathcal M}^{\rm disc}_{G',0,d;\kappa}(\mathbb{CP}^1,0)]^{\rm virt}.$$

The formula is proved in genus $0$ by the result cited in the source, while its validity in higher genus is not established here.