Gap Graphs of Numerical Semigroups

It is well known that if $S$ and $T$ are numerical semigroups, and its gap posets (with respect to the order induced by the semigroups) are isomorphic, then $S$ and $T$ must be the same.

Instead of the poset associated to the gaps of $S$, we can construct the gap graph associated to $S$ in the following way: its set of vertices are the gaps of $S$, and two gaps $h$ and $g$ ($h\neq g$) are joined by an edge if either $g-h\in S$ or $h-g\in S$.

So, it is natural to ask the same question holds for the gap graphs of $S$ and $T$: can we find two different numerical semigroups with isomorphic gap graphs?

Some tools to study this question are available in the repository