The field-of-definition conjecture for special subvarieties of variations of Hodge structure

Let $\mathbb V$ be a $\mathbb Z$-variation of Hodge structure defined over a number field $L\subset\mathbb C$. Let $\mathbb V^\otimes=\bigoplus_{a,b\in\mathbb N}\mathbb V^{\otimes a}\otimes(\mathbb V^\vee)^{\otimes b}$, and let $\operatorname{Hod}(\mathcal V^\otimes)$ and $\operatorname{HL}(S,\mathbb V^\otimes)$ denote the corresponding Hodge-tensor and tensorial Hodge loci. Field-of-definition conjecture. (a) Any special subvariety of $\mathcal V^\otimes$, respectively $S$, for $\mathbb V$ is defined over a finite extension of $L$. (b) Any of the finitely many $\operatorname{Gal}(\overline{\mathbb Q}/L)$-conjugates of a special subvariety of $\mathcal V^\otimes$, respectively $S$, for $\mathbb V$ is again a special subvariety of $\mathcal V^\otimes$, respectively $S$, for $\mathbb V$. The conjecture extends the corresponding expectation from geometric variations to arbitrary integral variations of Hodge structure; its geometric analogue is implied by the Hodge conjecture, while the general case is left as an expectation in the source.