Squarefree polynomials with prescribed coefficients

For nonempty subsets S0,…,Sn−1 of a (large enough) finite field F satisfying|S1|,…,|Sn−1|>2or|S1|,|Sn−1|>n−1, we show that there exist a0∈S0,…,an−1∈Sn−1 such thatTn+an−1Tn−1+…+a1T+a0∈F[T] is a squarefree polynomial.