  Recently, two of us have found a family of singularity-free rotating black
hole solutions in Einstein's conformal gravity. These spacetimes are
characterized by three parameters: the black hole mass $M$, the black hole spin
angular momentum $J$, and a parameter $L$ that is not specified by the theory
but can be expected to be proportional to the black hole mass $M$. The Kerr
black hole solution of Einstein's gravity is recovered for $L = 0$. In a
previous paper, we showed that X-ray data of astrophysical black holes require
$L/M < 1.2$. In the present paper, we report the results of a more
sophisticated analysis. We apply the X-ray reflection model {\sc relxill\_nk}
to \textsl{NuSTAR} and \textsl{Swift} data of the supermassive black hole in
1H0707-495. We find the constraint $L/M < 0.45$ (90% confidence level).
