Bena et al.'s tadpole conjecture

Let $X$ be a smooth fourfold with primitive integral Hodge class $G\in H_{2,2;\mathrm{prim}}(X)\cap H_4(X,\mathbb{Z})$. Suppose that a nonzero integer multiple $aG$ is a general Hodge cycle, meaning

$$aG=\sum_{i=0}^s n_i[Z_i],$$

for some $s\in\mathbb{N}$ and algebraic cycles $Z_i$. For each point $T$ in the moduli space, let $Q_T$ denote the tadpole pairing and let $V_{aG}$ be the Hodge locus associated to $aG$.

Tadpole conjecture. The tadpole contribution satisfies

$$Q_T(aG,aG)\gtrsim\frac{\chi(X_T)}{24}.$$

Moreover, there exists a constant $C>0$, depending a priori on the family, such that

$$\operatorname{codim}_B\bigl(V_{aG}\bigr)\leq C\cdot Q(aG,aG).$$

This is a heuristic formulation of the tadpole conjecture, motivated by the expectation that stabilizing many complex-structure moduli requires a large tadpole charge. The supplied text does not establish the claim or give evidence resolving it; its status is therefore left open.