The endpoints do not determine intermediate pages

Let $A$ be the one-generator complex ${\mathbb{F}}_2\{z\}$ in chain degree $0$, with $dz=0$ and filtration degree $0$. Let $B$ have generators $x$ in degree $1$ and $y,z$ in degree $0$, differential $dx=y$, $dy=dz=0$, and filtration $F_{0}B={\mathbb{F}}_2\{y,z\}$, $F_{1}B=B$, $F_{-1}B=0$. In both complexes $d$ preserves the filtration, so [F1] applies.

In $A$, $E_0=E_1={\mathbb{F}}_2\{z\}$ because the differential is zero. In $B$, the associated graded has basis $z,y$ in filtration $0$ and $x$ in filtration $1$; its $d_0$ is zero because $dx=y$ strictly lowers filtration. Therefore $E_1(B)$ has dimension $3$. The $d_1$ differential sends the class of $x$ to the class of $y$ and sends $z$ to zero, so $E_2(B)={\mathbb{F}}_2\{z\}$; likewise $E_2(A)={\mathbb{F}}_2\{z\}$.

The total homology of $A$ is ${\mathbb{F}}_2\{z\}$. In $B$, the pair $x\mapsto y$ is contractible and the remaining class is $z$, so $H(B)\cong {\mathbb{F}}_2\{z\}\cong H(A)$. The pair by itself has zero total homology but nonzero $E_1={\mathbb{F}}_2\{x,y\}$, with its $d_1$ killing both classes.

By step 2.1, $\dim E_1(A)=1$ while $\dim E_1(B)=3$, so their $E_1$ pages cannot be isomorphic even though their total homology groups agree by step 2.2. Convergence [F3] identifies only the limiting page with the associated graded of the shared total homology; it does not determine the earlier pages. Thus the stated endpoint-only claim is false.