  Conformal loop quantum gravity provides an approach to loop quantization
through an underlying conformal structure i.e. conformally equivalent class of
metrics. The property that general relativity itself has no conformal
invariance is reinstated with a constrained scalar field setting the physical
scale. Conformally equivalent metrics have recently been shown to be amenable
to loop quantization including matter coupling. It has been suggested that
conformal geometry may provide an extended symmetry to allow a reformulated
Immirzi parameter necessary for loop quantization to behave like an arbitrary
group parameter that requires no further fixing as its present standard form
does. Here, we find that this can be naturally realized via conformal frame
transformations in scalar-tensor gravity. Such a theory generally incorporates
a dynamical scalar gravitational field and reduces to general relativity when
the scalar field becomes a pure gauge. In particular, we introduce a conformal
Einstein frame in which loop quantization is implemented. We then discuss how
different Immirzi parameters under this description may be related by conformal
frame transformations and yet share the same quantization having, for example,
the same area gaps, modulated by the scalar gravitational field.
