Does there exist, for all $r\geq 2$, a basis $A$ of order $r$ (so that $f_r(n)>0$ for all large $n$) such that $$\sum_{n\leq x}f_r(n)^2 \ll x$$ for all $x$?

Erdős and Rényi proved by the probabilistic method that there exists a set $A$ such that $$\sum_{n\leq x}f_r(n)^2 \ll x$$ and $$\lvert A\cap [1,x]\rvert\gg x^{1/r}$$ for all $x$.