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It is shown that the numbers $c_i$ of chains of length $i$ in the proper part $L\setminus\{0,1\}$ of a distributive  lattice $L$ of length $\ell +2$ satisfy the inequalities $$c_0  \ldotsu003ec_{\ell}.$$ This proves 75% of the inequalities implied by the  Neggers unimodality conjecture.

Chain Polynomials of Distributive Lattices are 75% Unimodal