On the entrywise powers of matrices

If A is an n x n complex matrix and x is an element of C-n, the conjecture is that if we take the kth power of each component of Ax, the resulting vector belongs to the range of the matrix obtained by taking the kth power of the entries of AA*, where A* is the adjoint of A. The conjecture is proved here for any k greater than or equal to 2 when we add assumptions of either low dimension (namely, n less than or equal to 4) or low corank (0, 1, and, with some technical restrictions, 2). This problem arises in the study of the Jacobian Conjecture.