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When $\\lambda $ has four parts, we show that $F_{\\lambda}$ is unimodal when $\\lambda =\\langle n,n-1,n-2,n-3 \\rangle$, for any $n\\ge 4$, and that unimodality fails for the doubly-indexed, infinite family of partitions of the form $\\lambda=\\langle n,n-t,n-2t,n-3t \\rangle$, for any given $t\\ge 2$ and $n$ large enough with respect to $t$.\n  When $\\lambda $ has $b\\le 3$ parts, we show that our rank-generating functions $F_{\\lambda}$ are all unimodal. 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