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For $n\geq 6$ we provide a counterexample to the conjecture that every integral vector of a $n$-dimensional integral polyhedral pointed cone $C$ can be written as a nonnegative integral combination of at most $n$ elements of the Hilbert basis of $C$. In fact, we show that in general at least $\lfloor 7/6 \cdot n \rfloor$ elements of the Hilbert basis are needed.