Characterisation of the top K-polynomial on the moduli space of curves

Let $g>1$, let $\pi:\overline{\mathcal M}_{g,n}\to\overline{\mathcal M}_{g,0}$ be the morphism forgetting all marked points, and let $\psi_i$ be the cotangent-line classes at the marked points. Define $K_m$ by

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

where

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

Top K-polynomial conjecture. One has

$$K_{2g-2+n}=\left(\prod_{i=1}^n\psi_i\right)\pi^*K_{2g-2},$$

and $K_{2g-2}$ is, up to scale, the unique degree $2g-2$ polynomial with the property that all its pullbacks multiplied by products of the $\psi$ classes are polynomials in kappa classes. This would characterise the top K-polynomials and provide a mechanism for the proposed kappa-class vanishing relations; the statement is not established in the source.