First, I'm using the settings of https://en.wikipedia.org/wiki/NTRUEncrypt, with $L_f$ set of polynomials with $d_f+1$ coefficients equal to 1, $d_f$ equal to $-1$ and the remaining $N-2d_f-1$ equal to 0; and $L_g$ the set of polynomials with $d_g$ coefficients equal to 1, $d_g$ equal to $-1$ and the remaining $N-2d_g$ equal to 0. The natural numbers $d_f$ and $d_g$ are just fixed parameters of the scheme.
Suppose one receives a polynomial $h$ in the ring $R_{N,q}=\mathbb{Z}_q[X]/\langle X^N-1 \rangle$.
Question: Is it possible to determine if $h$ is a public-key, that is, is it possible to determine if $h$ is of the form $pf_q \cdot g \pmod{q}$?
My attempt: The NTRU hardness assumption says that from $h$ one cannot determine $f$ or $g$, otherwise the scheme would be useless. Although I couldn't answer my question, I came up with a test. Since $g(1)=0$, we must have $h(1)=0$. Hence if $h(1) \neq 0$ then $h$ is not a public-key. What can we test more?
PS: No zero-knowledge proof or similar things from the source of $h$ are given.