Grimm–Monnée complexity conjecture for atypical special subvarieties

Let $(\mathbb{V},Q)$ be a variation of polarized integral Hodge structures of even weight $2k$ on a smooth quasi-projective variety $S$. Let $\mathcal{A}_{0}(1,q)$ denote the degree-weighted collection of maximal atypical special subvarieties of zero period dimension, where the defining tensor has tensor degree at most $1$ and quadratic norm at most $q$. Suppose that the level of $\mathbb{V}$ is at least three. Grimm–Monnée conjecture. For any $\varepsilon>0$,

$$\#\mathcal{A}_{0}(1,q)=O(q^{\varepsilon}).$$

The conjecture concerns quantitative finiteness for atypical special subvarieties. The paper proves a related estimate for isolated maximal atypical special subvarieties under a $\mathbb{Q}$-simple adjoint Mumford–Tate-group hypothesis, while the supplied text gives no resolution of this broader conjecture.