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Let $f:X \\r S$ be a dominant morphism defined over $k$ from a smooth projective variety $X$ to a smooth projective variety $S$ of dimension $\\leq 2$ such that the general fibre of $f_\\Omega$ has trivial Chow group of zero-cycles. For example, $X$ could be the total space of a two-dimensional family of varieties whose general member is rationally connected. Suppose that $X$ has dimension $\\leq 4$. 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