Faithfulness of regular Stanley sequences to their cores

Let $S(A)=\{a_n\}$ and $S(A')=\{a'_n\}$ be Stanley sequences. Say that $S(A)$ is faithful to $S(A')$ if, for every $a'_n<\omega(A')$, there is an index $m$ such that $a_m=a'_n$. For a regular sequence, its core is denoted $S'(A)$. Faithfulness conjecture. Every regular Stanley sequence is faithful to its core. This proposed property would describe how the initial portion of a regular sequence retains the elements of its core; the source gives no proof or resolution.