Monotonicity of the number of functions determined by additive relations

Let $E$ be a set and $S$ a set of arithmetic functions. For positive integers $k$ and $ell$, let $n(E,S,k)$ denote the number of functions in $S$ determined by the relation

$$f(a_1+a_2+\dotsb+a_k)=f(a_1)+f(a_2)+\dotsb+f(a_k)$$

for arbitrary $a_i\in E$. The monotonicity conjecture for $n(E,S,k)$. If $E$ and $S$ determine $n(E,S,k)$ in this way, then

$$n(E,S,k)\ge n(E,S,\ell)$$

when $k\le \ell$. The paper describes this as a generalization of the preceding conjecture, and gives no resolution.