The vanishing conjecture for intersection numbers on the perfect cone compactification

Let $G:=g(g+1)/2$ be the dimension of the perfect cone compactification ${{\mathcal A}_g^{\operatorname{Perf}}}$, let $L$ be the determinant of the Hodge bundle, and let $D$ be its boundary divisor. Define

$$a_N^{(g)}:=\langle L^{G-N}D^N\rangle_{{\mathcal A}_g^{\operatorname{Perf}}}.$$

Vanishing conjecture. The intersection numbers $a_N^{(g)}$ for any $N$ vanish unless $G-N=k(k+1)/2$ for some $k\le g$. The claim extends the vanishing pattern established in the paper for $N<3g-3$ and predicts that nonzero top intersection numbers occur only when the power of $L$ equals the dimension of some $\mathcal A_k$.