If a case of the Bunyakovsky conjecture holds for degree-$n$ $p(x)$, does it hold for all degree-$n$ $f(x)$ satisfying the criterion?

Question

The Bunyakovsky conjecture states that if a polynomial $f(x)$ satisfies:

1. The leading coefficient is positive
2. The polynomial is irreducible over $\mathbb{Z}$
3. $f(1), f(2) \dots$ share no common factor (the coefficients of $f(x)$ should be relatively prime)

then the sequence $f(1), f(2) \dots$ contains infinitely many primes.

My question: if Bunyakovsky's conjecture does hold for a polynomial $p(x)$ of degree $n>1$ satisfying the above conditions, does that tell us anything about other polynomials of degree $n$ satisfying the above conditions? That is, if the conjecture holds true for a degree-$n$ polynomial, must it hold true for all other sufficient degree-$n$ polynomials? (I have no substantial reason to think it might, but I also have no substantial reason to think it might not, and it doesn't seem that far-fetched.)

Answer 1

If it holds for $f(x)$, then it also holds for $f(x+k)$ for any integer $k$. If $f(x/m)$ also has integer coefficients, then it also holds for $f(x/m)$. But I see no obvious reason (except the truth of the conjecture) why it should hold for arbitrary polynomials of the same degree satisfying the conditions.