Bounded-offset conjecture for digit sums of multiples

Let $b,r$ be positive integers with $\gcd(b-1,r)=1$, and let $c$ be any integer. Write $s_b(n)$ for the sum of the base-$b$ digits of $n$. Bounded-offset conjecture. There exists a constant $C$, depending only on $b$ and $r$, such that for every $k\geq 1$ there is an integer $n\leq k+C$ satisfying

$$s_b(kn)\equiv c\pmod r.$$

Moreover, one can take $C\leq b^{r+c}$. This conjecture proposes that the weak bound from Proposition can be replaced by a bound only a constant distance above $k$, generalizing the analogous phenomenon for the classical Thue–Morse sequence.