  We show that the strongest form of Hopkins' chromatic splitting conjecture,
as stated by Hovey, cannot hold at chromatic level n=2 at the prime p=2. More
precisely, for V(0) the mod 2 Moore spectrum, we prove that the kth homotopy
group of L_1L_{K(2)}V(0) is not zero when k is congruent to -3 modulo 8. We
explain how this contradicts the decomposition of L_1L_{K(2)}S predicted by the
chromatic splitting conjecture.
