Witten–Kontsevich conjecture on combinatorial classes and tautological classes

Let $\mathcal{M}_{g,P}$ be the moduli space of smooth genus-$g$ curves with marked points $P$, and let $W_{m_*,P}$ denote the combinatorial class associated with ribbon graphs having $m_i$ vertices of valency $2i+3$, where $m_*=(0,m_0,m_1,\dots)$ satisfies

$$\sum_i (2i+1)m_i=4g-4+2|P|.$$

Consider the polynomial algebra $\mathbb{Q}[t]=\mathbb{Q}[t_1,t_2,\dots]$, with each $t_i$ of degree $1$. Witten–Kontsevich conjecture. For every such $m_*$, there exists a polynomial $f_{m_*}in \mathbb{Q}[t]$ of degree $\sum_{igeq 1}m_i$ such that

$$W^*_{m_*,P}=f_{m_*}(\kappa_1,\kappa_2,\dots)in H^*(\mathcal{M}_{g,P}),$$

where $W^*_{m_*,P}$ is the Poincare dual of $W_{m_*,P}$. Moreover,

$$f_{m_*}(t)=\prod_{igeq 1}\frac{(2^{i+1}(2i+1)!! )^{m_i}}{m_i!}t_i^{m_i}+\text{(terms of lower degree)}.$$

This conjecture predicts that the combinatorial cycles arising from prescribed ribbon-graph valencies are tautological. Penner proved the case $W_5=12\kappa_1$, while Arbarello and Cornalba computed further cases and found strong evidence; the general assertion remains open in the supplied source.