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Erd\\H{o}s-Granville-Pomerance-Spiro conjectured that for any set $\\mathcal{A}$ of asymptotic density zero, the preimage set $s^{-1}(\\mathcal{A})$ also has density zero. We prove a weak form of this conjecture: If $\\epsilon(x)$ is any function tending to $0$ as $x\\to\\infty$, and $\\mathcal{A}$ is a set of integers of cardinality at most $x^{\\frac12+\\epsilon(x)}$, then the number of integers $n\\le x$ with $s(n) \\in \\mathcal{A}$ is $o(x)$, as $x\\to\\infty$. 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