Berry cancellation conjecture for Lipschitz–Killing curvatures

Let $T$ be a Gaussian eigenfunction on a compact manifold $\mathbb M$. For $u\in\mathbb R$, define the excursion set

$$A_u(T,\mathbb M):=\{x\in\mathbb M:T(x)>u\}.$$

For each Lipschitz–Killing curvature of $A_u(T,\mathbb M)$, consider its projection onto the second-order Wiener chaos. Berry cancellation conjecture. For excursion sets of Gaussian eigenfunctions on compact manifolds, the projection of each Lipschitz–Killing curvature onto the second-order Wiener chaos vanishes if and only if $u=0$. This would provide a major step toward a global understanding of Berry's cancellation phenomenon. In dimension two, the relevant curvatures are the area, half the boundary length, and the Euler–Poincaré characteristic; results for the two-sphere support the conjecture for the area and Euler–Poincaré characteristic, with boundary-length evidence also discussed, but the general statement remains open.