Is there an analogue of Alexander's Theorem for Brunnian braids/links?

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The Alexander's Theorem says that every knot or link can be represented as a closed braid. Is it true that every Brunnian link can be represented as a closed Brunnian Braid?

My guess is that the answer is No. Any counterexamples?

A braid is called Brunnian if removing any of its strings results in the trivial braid. Similarly a link is called Brunnian if removing any of its components results in the trivial link.