Gross-Hacking-Keel-Kontsevich (https://arxiv.org/abs/1411.1394) constructed a canonical basis (the so-called “theta basis”) for a cluster algebra, at least assuming it satisfies a certain combinatorial condition. This condition holds for the homogeneous coordinate ring of the Grassmannian (see e.g. https://arxiv.org/abs/1712.00447 or https://arxiv.org/abs/1803.06901). On the other hand, the $m$th homogeneous component of the coordinate ring is isomorphic to the $GL$ representation $V(m \omega)^*$ for the appropriate minuscule weight $\omega$, hence has a basis (the dual canonical basis of Lusztig/Kashiwara) coming from the theory of quantized enveloping algebras.

Could these bases be the same? There are some vague statements alluding to this possibility in GHKK, but nothing definitive.

I know the two bases share some interesting properties: e.g., the twisted cyclic shift symmetry of the Grassmannian permutes both bases according to promotion of semistandard tableaux (for the dual canonical basis this was proved by Rhoades, see https://arxiv.org/abs/1809.04965; for the theta basis this was proved by Shen and Weng https://arxiv.org/abs/1803.06901). Most naive bases (e.g. the standard monomial basis) do not satisfy this property.