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Around 2007, Zagier conjectured that if the rank $r$ Nahm sum for $(A,B,C)$ is modular, then so is the dual Nahm sum associated with $(A^{-1},A^{-1}B,B^\\mathrm{T} A^{-1}B/2-{r}/{24}-C)$. We construct some explicit rank four Nahm sums which are modular while their duals are not modular. 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