Hodge-class formula for the top J-polynomial on the moduli space of curves

Let $\overline{\mathcal M}_{g,n}$ be the moduli space of stable curves, let $\kappa_i$ be the kappa classes, and define $J_m$ by

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

where the coefficients $\sigma_i$ are determined by

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

Hodge formula conjecture. For $n=0$ or $n=1$, the top polynomial is

$$J_{2g-2+n}=\begin{cases}(-1)^g\lambda_{g-2}\lambda_g\in H^*(\overline{\mathcal M}_{g},\mathbb{Q}),&n=0,\\(-1)^{g-1}\lambda_{g-1}\lambda_g\in H^*(\overline{\mathcal M}_{g,1},\mathbb{Q}),&n=1. \end{cases}$$

Here $\lambda_i$ denotes the $i$th Chern class of the Hodge bundle. This has been checked up to genus $4$ using computational tools, but remains conjectural in general.