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Closed ideals of the algebra of absolutely convergent Taylor series

Let $Γ$ be the unit circle, $A(Γ)$ the Wiener algebra of continuous functions whose series of Fourier coefficients are absolutely convergent, and $A^+$ the subalgebra of $A(Γ)$ of functions whose negative coefficients are zero. If $I$ is a closed ideal of $A^+$, we denote by $S_I$ the greatest common divisor of the inner factors of the nonzero elements of $I$ and by $I^A$ the closed ideal generated by $I$ in $A(Γ)$. It was conjectured that the equality $I^A= S_I H^\infty \cap I^A$ holds for every closed ideal $I$. We exhibit a large class $\scr F$ of perfect subsets of $Γ$, including the triadic Cantor set, such that the above equality holds whenever $h(I)\capΓ\in\scr F$. We also give counterexamples to the conjecture.