  After its reduction by a gauge-fixing procedure, the family of linearly
polarized Gowdy $T^3$ cosmologies admit a scalar field description whose
evolution is governed by a Klein-Gordon type equation in a flat background in
1+1 dimensions with the spatial topology of $S^1$, though in the presence of a
time-dependent potential. The model is still subject to a homogeneous
constraint, which generates $S^1$-translations. Recently, a Fock quantization
of this scalar field was introduced and shown to be unique under the
requirements of unitarity of the dynamics and invariance under the gauge group
of $S^1$-translations. In this work, we extend and complete this uniqueness
result by considering other possible scalar field descriptions, resulting from
reasonable field reparameterizations of the induced metric of the reduced
model. In the reduced phase space, these alternate descriptions can be obtained
by means of a time-dependent scaling of the field, the inverse scaling of its
canonical momentum, and the possible addition of a time-dependent, linear
contribution of the field to this momentum. Demanding again unitarity of the
field dynamics and invariance under the gauge group, we prove that the
alternate canonical pairs of fieldlike variables admit a Fock representation if
and only if the scaling of the field is constant in time. In this case, there
exists essentially a unique Fock representation, provided by the quantization
constructed by Corichi, Cortez, and Mena Marugan. In particular, our analysis
shows that the scalar field description proposed by Pierri does not admit a
Fock quantization with the above unitarity and invariance properties.
