The conjectural formula for Kontsevich cycles $[W_{n,k}^\ast]$

Let $W_{n,k}^\ast$ be the Kontsevich cycle indexed by nonnegative integers $n$ and $k$, and let $\widetilde{\kappa}_j$ denote the corresponding modified kappa class. Write $(2m+1)!!$ for the odd double factorial. The conjectural formula for $[W_{n,k}^\ast]$. One should have

$$[W_{n,k}^\ast]=(-2)^{n+k+2}(2n+1)!!(2k+1)!!(\widetilde{\kappa}_n\widetilde{\kappa}_k-\widetilde{\kappa}_{n+k})-(-2)^{n+k+1}(2n+2k+3)!!\widetilde{\kappa}_{n+k}.$$

When $n=k$, the right-hand side should be divided by $2$. This conjecture extrapolates the formulas established for the cases $k=0$ and $k=1$ and predicts a symmetric expression for all Kontsevich cycles; its general validity is not established in the supplied text.