Lower-bound conjecture for minimal presentations of embedding-dimension-four numerical semigroups

Prove that every numerical semigroup $S$ with embedding dimension four satisfies $\eta(S)\geq t(S)+1$, where $t(S)$ is its type and $\eta(S)$ is the cardinality of a minimal presentation.

Background

The paper proves the general bounds $4t(S)+5\geq \eta(S)\geq t(S)-11$ for numerical semigroups of embedding dimension four. The authors state that the lower bound can most likely be improved and formulate the stronger inequality $\eta(S)\geq t(S)+1$ as a conjecture.

The conjecture is motivated by the absence of known embedding-dimension-four examples with $\eta(S)\leq t(S)$ and by several examples attaining equality $\eta(S)=t(S)+1$. The authors explicitly qualify the proposed inequality as conjectural rather than established.