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