  In this note we prove that the Bass stable rank of $H^\infty_\R(\D)$ is two.
This establishes the validity of a conjecture by S. Treil. We accomplish this
in two different ways, one by giving a direct proof, and the other, by first
showing that the topological stable rank of $H^\infty_\R(\D)$ is two. We apply
these results to give new proofs of results by R. Rupp and A. Sasane stating
that the Bass stable rank of $A_\R(\D)$ is two and the topological stable rank
of $A_\R(\D)$ is two, settling a conjecture by the second author. We also
present a $\bar\partial$-free proof of the second author's characterization of
the reducible pairs in $A_\R(\D)$.
