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Assume the Leray spectral sequence of a map $f_i:X_i\rightarrow B_i$ $E_2$-degenerates for $i=1,2$. Is it true that the Leray spectral sequence of the map $f_1\times f_2:X_1 \times X_2 \rightarrow B_1 \times B_2$ $E_2$-degenerates as well?

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