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Hello and unfortunately no. If the will appointed the public trustee as the executor then they will have to handle the probate on the will.
Be aware though that being the executor does not allow them to vary the distribution set out in the will, it merely means they have to supervise the process until a grant of probate is made and must then distribute the estate in accordance with the will.
I appreciate this is not the answer you were likely hoping for but I trust the above assists your understanding.
Good luck and please rate my answer.