Generalized tadpole conjecture for the Hodge locus

Let $E\rightarrow\mathcal{M}$ be a variation of polarized Hodge structure of weight $D=2k$. For a positive integer $L$, define

$$E_{\mathrm{Hodge}}(L)=\{(z^i,v)\in E:v\in H^{k,k}\cap H_{\mathbb{Z}},\ (v,v)\leq L\}.$$

Here $\mathcal{M}$ is the complex-structure moduli space. Generalized tadpole conjecture. There exist positive constants $C_1,C_2$, independent of $L$ and $\operatorname{dim}\mathcal{M}$, such that if

$$\operatorname{dim}\mathcal{M}>C_1\qquad\text{and}\qquad \operatorname{dim}\mathcal{M}>C_2L,$$

then every connected component of $E_{\mathrm{Hodge}}(L)$ has strictly positive dimension. If the variation comes from the middle cohomology of a family of Calabi--Yau fourfolds, $C_2$ is expected to be of order one. This formalizes the tadpole expectation that fully stabilized vacua require a tadpole growing at least linearly with the dimension of moduli space. The source does not state a resolution, so the conjecture remains open.