The generalized periodic characterization conjecture for S_{1,g,g+1}

For $g\geq 2$, let $S_{1,g,g+1}$ be the greedy 3-subsumfree sequence beginning with $1,g,g+1$, where each subsequent term is the smallest larger integer that is not the sum of three distinct previous terms. The generalized periodic characterization conjecture. For every $g\geq2$,

$$z\in S_{1,g,g+1}\iff z\in\{1,2g+1,6g+1\}\ \text{or}\ z\bmod(10g+3)\in\{g,g+1,\ldots,2g\}\cup\{6g+2,6g+3,\ldots,7g+1\}.$$

In particular, after the first $g+4$ entries, the sequence modulo $10g+3$ is periodic with period $2g+1$. The paper reports verification only for $g\in\{2,3,\ldots,10\}$ and leaves the parametrized statement conjectural.