Conjecture on the asymptotic error term for floor function sums over powers

Let $r\geq 2$ be an integer. Let $f$ be given by the paper's definition in, with $h$ satisfying $|h(d)|\ll d^\alpha$ for some $\alpha\in[0,\frac{1}{2r-1}]$, and let $C_f$ be the constant defined in. The asymptotic error-term conjecture. One should have

$$S_f(x)=C_f x+O\left(x^{\frac{1+\alpha}{2r}}\right).$$

This would be a simpler and stronger estimate than the preceding theorem in the indicated range of $\alpha$, improving the available error term for these floor-function sums over powers.