Orbit-closure conjecture for times two and three on tori

Let $d\geq 2$ be an integer. Let $A\subset\mathbb{T}^d$ be an orbit closure under the $\times 2,\times 3$ actions. Orbit-closure conjecture. Then $A$ is a union of finitely many (possibly non-homogeneous) subtori. This conjecture seeks a structural classification of orbit closures without the irreducibility hypothesis used in earlier invariant-set results. The surrounding discussion notes that the Hausdorff dimension of a closed invariant set is restricted to an integer, but the one-dimensional case can still be complicated; the conjecture remains unresolved in the supplied text.