Nonnegativity of the beta parameter for regular Stanley sequences

Let $S(A)=\{a_n\}$ be a regular Stanley sequence. For sufficiently large $k$, write

$$a_{2^k-\sigma(A)}=\alpha(A)3^k+\beta(A)2^k,$$

where $\sigma(A)$ is the shift index and $\beta(A)$ is the associated rational parameter. Nonnegativity conjecture.

$$\beta(A)\ge 0.$$

The parameter $\beta(A)$ controls the secondary term in the growth of a regular Stanley sequence. Its nonnegativity would support the conclusion that all regular Stanley sequences have Type 1 growth; the source presents this as suggested by data.