The vanishing conjecture for Morita–Mumford classes of surface bundles

Let $B$ be the base of a surface bundle, let $p$ be a prime, and let $\kappa_i$ denote its Morita–Mumford classes. Vanishing conjecture. For $s\geq 1$ and $v\geq 0$, one has

$$\kappa_{p^v s(p-1)-1}=0\in H^*(B;\mathbb{Z}/p^{v+1}).$$

The theorem preceding this conjecture proves the case $v=1$. The statement is best possible: if $s\not\equiv0\pmod p$, then the corresponding class is nonzero with coefficients in $\mathbb{Z}/p^{v+2}$; hence the conjecture itself is solved, as indicated by the supplied status evidence.