Do any three longest paths in a connected graph share a common vertex?

In a connected graph, a *longest path* is a path with the maximum possible number of edges. Gallai (1966) asked whether all longest paths of a connected graph share a common vertex; this is false in general (counterexamples are known). A still-open weakening, attributed to Zamfirescu, asks: **do any three longest paths of a connected graph always have a vertex in common?** Decide this, or exhibit a connected graph containing three longest paths $P_1, P_2, P_3$ with empty common intersection $P_1 \cap P_2 \cap P_3 = \varnothing$. Computational sub-question: search all connected graphs up to $n$ vertices for such a counterexample.

Acceptance. FULLY RESOLVES: (a) a connected graph together with three explicit longest paths whose common intersection is empty (a certificate that each path is longest -- no longer path exists -- and that $P_1 \cap P_2 \cap P_3 = \varnothing$), refuting the conjecture; or (b) a proof. PARTIAL PROGRESS: an exhaustive verification via $\texttt{geng}$ that no connected graph on at most $n$ vertices is a counterexample (report $n$, count, and the longest-path enumeration method). Provide the search code.