Density conjecture for modular Stanley sequences

Let $S(0)=(s_n)$ be the Stanley sequence generated by $\{0\}$. A modular sequence $S(A)=(a_n)$ is denser than $S(0)$ eventually if its terms satisfy $a_n<s_n$ for all sufficiently large $n$.

Density conjecture. There exists a modular sequence $S(A)=(a_n)$ such that

$$a_n<s_n$$

for all sufficiently large $n$.

The paper exhibits a modular sequence with this inequality for infinitely many $n$, but no example satisfying it eventually is known there.