Let $R$ be an ungraded commutative ring and let $M$ and $N$ be $R$- modules equipped with increasing and exhaustive filtrations, i.e $\{0\}\subset F_0P\subset\dots \subset F_nP\subset\dots\subset P$ such that $P=\cup_iF_iP$ for $P=M, N$. Let $f: M\to N$ be a morphism of filtered modules, i.e. $f(F_nM)\subseteq F_nN$ for every $n$ such that the induced map $\mathrm{Gr}(f): \mathrm{Gr}_{\bullet}(M)\to\mathrm{Gr}_{\bullet}(N)$ is an isomorphism of graded $R$-modules. Is it true that $f: M\to N$ is an isomorphism of filtered $R$-modules?
My opinion is the following: Based on the proof of the Lemma in the second answer to the question Isomorphism of Associated Graded Algebras, I believe that the exhaustiveness of the filtrations of $M$ and $N$ are enough to prove the above claim with the help of the Five Lemma.
EDIT: I added the specification "isomorphism of graded $R$-modules" instead of just "isomorphism of $R$-modules".