Pseudomodularity conjecture for Type 1 Stanley sequences

Let $S(A)=(a_n)$ be a Stanley sequence. A sequence is pseudomodular if it has the shifted recursive structure defined using a modular core, character, repeat factor, and shift index.

Pseudomodularity conjecture. A Stanley sequence follows Type 1 growth if and only if it is pseudomodular.

Every regular Stanley sequence is pseudomodular, and the paper proves that every pseudomodular sequence has Type 1 growth. Thus the unresolved direction is that every Type 1 sequence should be pseudomodular.