  The loop quantization of the conformal Brans-Dicke cosmology is explored in
the spatially flat and Bianchi-I setting. The scalar and conformal constraints
governing the canonical model are quantized using the loop techniques. The
physical Hilbert space of quantum spacetimes satisfying both quantum
constraints is then obtained by incorporating the quantum geometric features.
The Schr\"odinger cosmic evolutions are derived with the relational Heisenberg
observables describing the dynamical degrees of freedom with respect to the
chosen reference degrees of freedom, with the latter providing the physical
coordinates for the spatial hypersurfaces and the conformal scales. We show
that the emerging Schr\"odinger theories contain not only the loop quantum
cosmology of GR, but also that of the so-called shape dynamics. The exact
dictionary between the two theories is achieved via the underlying physical
Hilbert space possessing the additional (loop-corrected) conformal symmetry.
