Prove monotonicity in the integer Rényi index

Prove that the Rényi multi-entropy S_n^{(q)} for stabilizer states is nonincreasing as an integer function of n for all integers n≥2.

Background

The paper observes that polynomial continuations of Rényi multi-entropies need not be monotone for real n>0, unlike ordinary Rényi entropy. For the directly defined integer-replica quantities, however, the examples studied in the paper appear to satisfy nonincreasing behavior.

The authors do not establish this property in general and explicitly identify a general proof of integer-n monotonicity as an open problem.