The CM-point characterization conjecture

Let $(G,X,\Gamma)$ be a real Shimura datum and let $W\subset S_\Gamma$ be an irreducible algebraic subvariety. A CM-point is a point of $S_\Gamma$ at which the Mumford–Tate group of the induced integral variation of Hodge structure is commutative. The CM-point characterization conjecture. The subvariety $W$ contains a Zariski dense set of CM-points if and only if it is special and arithmetic. The source identifies this as a special case of Klingler's conjecture and as a consequence predicted by the preceding $\mathbb{Z}$-special-point conjecture; it remains open there.