The stable-rank-two conjecture for the real disc algebra

Let $A_\mathbb{R}(\mathbb{D})$ denote the real disc algebra, consisting of functions analytic on $\mathbb{D}$, continuous on $\overline{\mathbb{D}}$, and satisfying the relevant real-symmetry condition. Its stable rank is the least integer controlling unimodular tuples in the algebra. Stable-rank-two conjecture. The stable rank of $A_\mathbb{R}(\mathbb{D})$ is two. The preceding argument establishes only that the stable rank is at least two; the conjecture asserts the matching upper bound and hence an exact value.