Density conjecture for modular $p$-Stanley sequences

Let $S_p(0)=(s_n)$ be the $p$-Stanley sequence generated by $\{0\}$. A modular $p$-Stanley sequence $S_p(A)=(a_n)$ is eventually denser than $S_p(0)$ when $a_n<s_n$ for all sufficiently large $n$.

$p$-density conjecture. There exists a modular $p$-Stanley sequence $S_p(A)=(a_n)$ such that

$$a_n<s_n$$

for all sufficiently large $n$.

This is presented as the $p$-free analogue of the density conjecture for ordinary modular Stanley sequences; no resolution is supplied.