What are the security implications of knowing the private polynomial $\mathcal{F}$
Let's answer my question after revisiting the research. Start by quoting the following text from [2]. Note that I've changed the variables names to fit my description. Here $P(X) = T \circ F \circ S(X)$.


  An open question is to know whether or not these schemes remain secure
  if the set of polynomials $\mathcal{F}$ is public. In this situation,
  the security of these schemes relies not only on the difficulty of
  finding a common zero of a system of non linear equations but also on
  the difficulty of $\mathcal{IP}$ (Isomorphism of Polynomials) problem
  (when $\mathcal{F}$ and $\mathcal{P}$ are given, the problem is to recover the
  pairs $(A_1,v_s)$ and $(A_2,v_t)$).


In this post the private polynomial $\mathcal{F}$ is public, moreover described methods reduce IP2 to IP1 if $T$ is known, and IP1 is totally insecure [2]. In the other hand, if $S$ is known $T$ is recovered by basic linear algebra.

Then I conclude that a cryptosystem where $\mathcal{F}$ is known depends on the IP2 assumption (finding $(S,T))$ and the $\mathcal{MQ}$ problem implicitly. You can follow up from [2] section $6.2$

EDIT: Moreover, it results that IP1 is totally broken when the private polynomial is known. However if the polynomial $\mathcal{F}$ is unknown, having access to an oracle then querying for canonical vectors as a plaintext will reveal $S$ too. A secure cryptosystem based on MPKC must satisfy the IP2 assumption from design, between others.