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We prove that, for any $f_1,\\ldots, f_k\\in L^{\\infty}(X)$, the M\\\"obius-weighted polynomial multiple ergodic averages \\begin{align*}\\frac{1}{N}\\sum_{n\\leq N}\\mu(n)f_1(T^{P_1(n)}x)\\cdots f_k(T^{P_k(n)}x) \\end{align*} converge to $0$ pointwise almost everywhere. Specialising to $P_1(y)=y, P_2(y)=2y$, this solves a problem of Frantzikinakis. 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