On chromatic polynomials and the golden ratio

Let M be a triangulation of the 2-sphere, with k vertices. Let P(M, n) be its chromatic polynomial with respect to vertex-colorings. Then$\left|P\right(M,1+\tau \left)\right|⩽{\tau }^{5-k}$where τ is the “golden ratio” $(1+\sqrt{5})/2$

This result is offered as a theoretical explanation of the empirical observation that P(M, n) tends to have a zero near n=1+τ (see [1]).