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They conjectured that, if $\\mathcal{F} \\subset \\binom{[n]}{k}$ and $\\mathcal{G} \\subset \\binom{[n]}{\\ell}$ are non-trivial cross-intersecting families with $n \\geq 2k > 2\\ell \\geq 4$, the maximum of $|\\mathcal{F}||\\mathcal{G}|$ is attained by the natural Hilton-Milner-type configurations.\n  In this paper, we present two main results concerning this conjecture. Firstly, we show that the conjecture does not hold in general. 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