Equality of the canonical-basis-positive and totally positive parts

Prove that the subspace of the coordinate algebra spanned by the dual canonical basis, O′≥0, equals the totally positive part O≥0; equivalently, determine whether the canonical basis B can be reconstructed from the totally positive semigroup U(R>0) by identifying B∗ with the indecomposable nonzero elements of O≥0.

Background

The paper defines O≥0 as the semiring of linear functionals on the completed enveloping algebra whose values on the totally positive semigroup U(R>0) are nonnegative real numbers. It also defines O′≥0 as the subspace of the coordinate algebra spanned by the dual canonical basis B∗. Proposition 0.7 establishes the inclusion O′≥0 ⊂ O≥0.

Section 0.8 conjectures that this inclusion is an equality. The conjecture is known in type A2, but the author notes that it may be too optimistic in general. An equivalent formulation asks whether the canonical basis can be recovered from U(R>0) as the set of elements of O≥0 that cannot be expressed as sums of two nonzero elements of O≥0.