If I understand correctly, a Fujisaki commitment is as follows: $g^m \cdot h^r $ mod $n$, where $m$ is a message, $r$ is a random number, there exists $a$ such that $h^a = g$, and $n$ is an RSA modulus.
Usually, when I read discussions about this scheme, the holder of the commitment does not know the prime factorization of $n$.
Is it a requirement that the order of the group be unknown to the creator of the commitment? (Note added assumption below.)
EDIT: Note that the message can be replaced with $m+k\cdot \lambda(n)$, where $\lambda(n)$ is the order of $n$. Though this is a threat to the commitment, I am more concerned with commitments of small values ($m << \lambda(n)$), so opening the commitment like this would be rejected. So I will rephrase:
Assuming $m << \lambda(n)$, is a Fujisaki Commitment binding if the prime factorization of $n$ is known?