Betti status of the boundary elements in secondary numerical semigroups

Determine whether the elements $b_{22}d_2$ and $b_{33}d_3$ are Betti elements for secondary numerical semigroups $S=\langle d_0,d_1,d_2,d_3\rangle$ with embedding dimension four.

Background

For secondary numerical semigroups, the paper identifies a large collection of Betti elements arising from $(2,2)$-type points and proves that each such element has exactly two $\mathcal R$-classes. It also establishes that $a_{ii}d_i$ is a Betti element for each generator index, but the corresponding analysis does not settle the status of the boundary elements $b_{22}d_2$ and $b_{33}d_3$.

The unresolved status does not affect the bounds proved later for the type and the cardinality of minimal presentations, because the authors obtain sufficiently strong upper and lower estimates without deciding whether these two elements are Betti elements.