Random ordering in modulus of consecutive Hecke eigenvalues of primitive forms

Let tau(.) be the Ramanujan tau-function, and let k be a positive integer such that tau(n) is not 0 for n=1,...,[k/2]. (This is known to be true for k < 10^{23}, and, conjecturally, for all k.) Further, let s be a permutation of the set {1,...,k}. Then there exist infinitely many positive integers m such that |tau(m+s(1))|<tau(m+s(2))|<...<|tau(m+s(k))|. We also obtain a similar result for Fourier-coefficients of general newforms.