  Weyl conformal symmetry can solve the problem the spacetime singularities
present in Einstein's gravity. In a recent paper, two of us have found a
singularity-free rotating black hole solution in conformal gravity. In addition
to the mass $M$ and the spin angular momentum $J$ of the black hole, the new
solution has a new parameter, $L$, which here we consider to be proportional to
the black hole mass. Since the solution is conformally equivalent to the Kerr
metric, photon trajectories are unchanged, while the structure of an accretion
disk around a black hole is affected by the value of the parameter $L$. In this
paper, we show that X-ray data of astrophysical black holes require $L/M <
1.2$.
