This short communication considers the LU factorization with partial pivoting and shows that an all-at-once result is possible for the structure prediction of the column dependencies in L and U. Specifically, we prove that for every square strong Hall matrix A there exists a permutation P such that every edge of its column elimination tree corresponds to a symbolic nonzero in the upper triangular factor U. In the symbolic sense, this resolves a conjecture of Gilbert and Ng [6].