Quadratic period relations and Shimura subvarieties for CM abelian varieties

Let $A$ be a CM abelian variety of dimension $g\geq 1$, let $[o]\in\mathbb{A}_g(\overline{\mathbb{Q}})$ parametrize $A$, and set $[\mathbf{o}]:=([o],\ldots,[o])$. An elementary quadratic relation is a non-trivial elementary algebraic relation of degree two between the holomorphic periods of $A$. A root space is a root space of the tangent space $T_{[o]}\mathbb{A}_g$. The Shimura-subvariety conjecture. The following are equivalent: (i) there exist non-trivial elementary quadratic relations between the holomorphic periods of $A$; (ii) there exist $r\geq 1$ and a Shimura subvariety $S$ of $\mathbb{A}_{gr}$ such that $S$ contains $[\mathbf{o}]$ and

$$T_{[\mathbf{o}]}S$$

is not the direct sum of root spaces of $T_{[o]}\mathbb{A}_g$. The equivalence would connect quadratic period relations with exceptional tangent subspaces on Shimura varieties; the supplied text gives no resolution, so the conjecture remains open.