Let $(C_{p,q},\partial,\delta)$ be a double complex in homological grading. By consider horizontal or vertical filtrations, one gets two spectral sequences.
It is well-known that if $C_{p,q}=0$ whenever $p<0$ or $q<0$, then both spectral sequences are first-quadrant, and both converge to the homology of $\mathrm{Tot}(C)$.
My question is: Suppose $C_{p,q}=0$ whenever $p<0$, but there could be $C_{p,q}\ne0$ for $q<0$. Then the spectral sequences are not first-quadrant, but are first-or-fourth-quadrant. If I assume $E^2_{p,q}$ become first-quadrant for both filtrations, so that they converge. Then do these two spectral sequences converge to the same limit? If so, do they converge to the homology of $\mathrm{Tot}^{\prod}(C)$?