Combinatorial freeness property for path-with-triangle arrangements

Determine whether the connected subgraph arrangements associated with path-with-triangle graphs satisfy a stronger combinatorial freeness property.

Background

The conjecture that freeness is equivalent to inductive freeness would imply a stronger combinatorial freeness result for the classified free families. The authors specifically identify the path-with-triangle family as unresolved, because it is not known whether its arrangements possess a stronger combinatorial freeness property.