Full upper-range saturation spectrum for Berge stars of size at least six

Determine the full upper range of the saturation spectrum for 3-uniform Berge-$K_{1,\ell}$ with every integer $\ell\ge 6$, without relying on the development of significant new tools.

Background

The paper completely determines the 3-uniform saturation spectrum of Berge-$K_{1,5}$ when $5$ divides the number of vertices, and determines all but a constant number of values for Berge-$K_{1,\ell}$ when $\ell\ge 5$. The unresolved difficulty concerns the upper range for $\ell>5$, where the difference between the edge counts of the $\ell$-lantern and three copies of the complete 3-uniform hypergraph on $\ell$ vertices grows with $\ell$.

For $\ell=5$, the authors can handle the finitely many possible deficits through seven cases. As $\ell$ increases, this case analysis becomes infeasible; the authors specifically identify determining the complete upper range for all $\ell\ge 6$ as unresolved. They note that the case $\ell=6$ might be approachable with substantial effort, but that a general solution would require significant new tools.