Hodge double ramification cycle conjecture

Let $g,k,u\geq 0$ and let $A=(a_1,\ldots,a_n)\in\mathbb{Z}^n$ satisfy

$$|A|=k(2g-2+n).$$

Let $\overline{{\mathcal M}}_{g,A}^{k,r}$ be the moduli space of twisted $r$-spin structures, let $\mathcal L$ be its universal line bundle, and let $\epsilon\colon\overline{{\mathcal M}}_{g,A}^{k,r}\to\overline{{\mathcal M}}_{g,n}$ be the natural morphism. Define

$${\rm Ch}_{g,A}^{k,r,d}=r^{2d-2g+1}\epsilon_*c_d(-R^*\pi_*\mathcal L)\in R^d(\overline{{\mathcal M}}_{g,n}).$$

Let ${\mathbf{Rub}}_{\mathcal L_A}$ denote the rubber space identified with the relevant moduli space of relative stable maps, let $p$ be its forgetful map to $\overline{{\mathcal M}}_{g,n}$, and let $\eta$ be its cotangent-line divisor class. Hodge DR conjecture. For every $g,k,u\geq 0$ and every $A\in\mathbb{Z}^n$ with $|A|=k(2g-2+n)$,

$$p_*\left(\left[\mathbb{P}\big({\mathbf{Rub}}_{\mathcal L_A}\big)\right]^\mathrm{vir}\cdot\eta^u\right)=[r^u]{\rm Ch}_{g,A}^{k,r,g+u}\in\mathrm{CH}^{g+u}(\overline{{\mathcal M}}_{g,n}),$$

where $[r^u]$ denotes the coefficient of $r^u$. This conjecture proposes a relation between virtual pushforwards from rubber relative stable-map spaces and coefficients of Chiodo's classes, connecting logarithmic and multi-scale differential moduli spaces; the relation remains unresolved in the source.