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For $t=1$, this problem is exactly equivalent to Czabarka's partition-EKR conjecture, first introduced in print by P.~L. Erd\\H{o}s and L.~A. Sz\\'ekely~\\cite{ErdosSzekelyHigher}. We prove the corresponding Erd\\H{o}s--Ko--Rado theorem in the explicit linear range $n+1\\ge8k$, giving a constant-factor advance toward the conjectured sharp range $n\\ge2k$. 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We study families $\\mathcal{A}\\subseteq\\mathcal{F}_k(M_n)$ satisfying $\\mathrm{rk}(A\\wedge B)\\ge t$ for all $A,B\\in\\mathcal{A}$. For $t=1$, this problem is exactly equivalent to Czabarka's partition-EKR conjecture, first introduced in print by P.~L. Erd\\H{o}s and L.~A. Sz\\'ekely~\\cite{ErdosSzekelyHigher}. We prove the corresponding Erd\\H{o}s--Ko--Rado theorem in the explicit linear range $n+1\\ge8k$, giving a constant-factor advance toward the conjectured sharp range $n\\ge2k$. 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