g.c.d. and Euler's totient function

I have made some computations which seem to corroborate the OP's conjecture, namely that for any $n$ there exists a $N$, such that for every polynomial $P$ of degree $n$, with positive integral coefficients and content 1, the quantity $$g(P):= g.c.d(\phi(P(x)),x \geq 1)$$ divides $N$.

For $n=1$, as the OP says, one can take $N=8$ as proved by Serre.

For $n=2$, it seems that one can take $N=2^4 3^2 = 144$. It seems even more that one cannot do better, because for $P(x)=16x^2+32x+17$, I get experimentally $g(P)=16$ (this must not be hard to prove but I haven't tried), and for $P(x)=27 x^2 + 9x+1$, I get $g(P)=18$. So $144 | N$. On the other hand I have need been able to find any $P$ such that $g(P)$ was not a divisor of $144$.

For $n=3$ or $n=4$, I have failed to find any $P$ with $g(P)\geq 2$. This suggests $N=2$ in these cases.