The Kleiman-Piene conjecture for nodal curves

Let $B$ be a family of curves satisfying the hypotheses ${\bf DIM}_{\mathrm{KP}}$: the locus of non-reduced curves has codimension greater than $\delta$, and for every equisingularity type $\mathbf{D}$, the locus $B(\mathbf{D})$ has either at least the expected codimension $\operatorname{cod}(\mathbf{D})$ or codimension greater than $\delta$. Let $B(\delta)$ denote the locus of $\delta$-nodal curves. Under these hypotheses, there is a natural non-negative cycle $U(\delta)$ supported on $\overline{B(\delta)}$. Kleiman-Piene conjecture. The locus $B(\delta)$ is either empty or has pure codimension $\delta$, and the rational equivalence class $[U(\delta)]$ is given by the universal polynomial in the classes $\epsilon(a,b,c)$ asserted by the Kleiman-Piene theorem for every $\delta\geq 0$. Kleiman and Piene proved this statement for $\delta\leq 8$; the conjecture extends their universal-polynomial formula and codimension assertion to all numbers of nodes.