Theta-class formula for the top K-polynomial on the moduli space of curves

For $g,n\in\mathbb{N}$ with $2g-2+n>0$, let $K_m$ be the homogeneous polynomials in the kappa classes defined by

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

where

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

and let $\Theta_{g,n}\in H^{2(2g-2+n)}(\overline{\mathcal M}_{g,n},\mathbb{Q})$ be the classes introduced in the cited construction. Theta-class conjecture. For $2g-2+n>0$,

$$K_{2g-2+n}=\Theta_{g,n}.$$

The classes $\Theta_{g,n}$ are conjecturally related to the cohomology of the supermoduli space of curves; this identity is proposed as another characterisation of the top K-polynomial and remains open.