The product of $E_2$-degenerate spectral sequences also $E_2$-degenerates?

Question

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?