Polynomial vanishing relations for kappa classes on the moduli space of stable curves

Let $\overline{\mathcal M}_{g,n}$ be the moduli space of genus $g$ stable curves with $n$ marked points, and let $\kappa_m\in H^{2m}(\overline{\mathcal M}_{g,n},\mathbb{Q})$ be the Miller–Morita–Mumford classes. Define homogeneous polynomials $K_m$ and $J_m$ by

$$\exp\left(-\sum_{i>0}s_it^i\right)=\sum_{k=0}^\infty(-1)^k(2k+1)!!t^k,$$ $$\exp\left(-\sum_{i>0}\sigma_it^i\right)=\sum_{k=0}^\infty(-1)^kk!t^k,$$

and

$$\sum_{m\geq 0}K_m=\exp\left(\sum_{i>0}s_i\kappa_i\right),\qquad \sum_{m\geq 0}J_m=\exp\left(\sum_{i>0}\sigma_i\kappa_i\right).$$

Kappa-class vanishing conjecture. The polynomial relations

$$K_m(\kappa_1,\ldots,\kappa_m)=0$$

for $m>2g-2+n$, except when $(m,n)=(3g-3,0)$, and

$$J_m(\kappa_1,\ldots,\kappa_m)=0$$

for $m>2g-2+n$, together with $m=2g-2+n$ when $n>1$, hold in the cohomology of $\overline{\mathcal M}_{g,n}$. These relations are proposed as polynomial relations among kappa classes on the compactified moduli space; their general validity is the subject of the paper, while related compact-type relations are known.