The generalized Füredi conjecture holds for finite linear lattices

We say that a rank-unimodal poset P has rapidly decreasing rank numbers, or the RDR property, if above (resp. below) the largest ranks of P  , the size of each level is at most half of the previous (resp. next) one. We show that a finite rank-unimodal, rank-symmetric, normalized matching, RDR poset of width ww has a partition into ww chains such that the sizes of the chains are one of two consecutive integers. In particular, there exists a partition of the linear lattices Ln(q)Ln(q) (subspaces of an n  -dimensional vector space over a finite field, ordered by inclusion) into chains such that the number of chains is the width of Ln(q)Ln(q) and the sizes of the chains are one of two consecutive integers.