  We argue that the Hausdorff dimension D of a quantum gravity random surface
is always D=4, irrespective of the conformal central charge c, c between -2 and
1, of a critical statistical model possibly borne by it. The
Knizhnik-Polyakov-Zamolodchikov (KPZ) relation allows one to determine the
dimension D from the exact Hausdorff dimension of random maps with large faces,
recently rigorously studied by Le Gall and Miermont, and reformulated by Borot,
Bouttier and Guitter as a loop model on a random quadrangulation. This
contradicts several earlier conjectured formulae for D in terms of c.
