An early (and influential) scaling relation in the multifractal theory of diffusion limited aggregation (DLA) is the Turkevich-Scher conjecture that relates the exponent [Formula presented] that characterizes the “hottest” region of the harmonic measure and the fractal dimension D of the cluster, i.e., [Formula presented] Due to lack of accurate direct measurements of both D and [Formula presented] this conjecture could never be put to a serious test. Using the method of iterated conformal maps, D was recently determined as [Formula presented] In this paper, we determine [Formula presented] accurately with the result [Formula presented] We thus conclude that the Turkevich-Scher conjecture is incorrect for DLA.