Conformal scalar and torsion do not go together

We investigate the conformal transformation of vierbein-Einstein-Palatini (VEP) action in terms of tetrads $e^I_\mu$ and spin connection $A^{IJ}_\mu$. The transformation of the spin connection is indeterminate off-shell unless equations of motion are satisfied. We construct the conformally invariant scalar field in the torsion-free VEP formalism. In presence of fermionic matter, torsion does not vanish, and shows up in the dynamics of conformal scalar, affecting the invariance. It is not possible to maintain conformal invariance of the scalar field equation when fermions are present.