Do roots of a polynomial with coefficients from a Collatz sequence all fall in a disk of radius 1.5?

Question

Consider a modified version of Collatz sequence:

$C(n)=\left\{ \begin{array}{ll} \frac{3n+1}{2} & n\ \mathrm{odd} \\ \frac{n}{2}& n\ \mathrm{even}\end{array} \right.$

Let $F_n$ be the smallest integer that satisfies $C^{F_n}(n)=1$, where $C^0(n)=n$, $C^1(n)=C(n)$, and $C^k(n) = C(C^{k-1}(n))$ for any integer $k>1$.

Now for any integer $n>0$ with a finite $F_n$ (can be infinite if Collatz conjecture is disproved) consider polynomial

$T_n(z)=n z^{F_n} + C(n) z^{F_n-1} + C^2(n) z^{F_n-2} + \cdots + C^{F_n-1} z + 1 = \sum_{k=0}^{F_n} C^k(n) z^{F_n-k}$

with roots $r^{(n)}_1, r^{(n)}_2, \ldots, r^{(n)}_{F_n}$.

Question: Can one prove that for any $n\neq9$, all roots are contained in an open disk of radius 1.5 around $(0,0)$, i.e.

$\forall 1\leq k \leq F_n, |r^{(n)}_k|<1.5$.

Note that $T_9(z)$ has a root at $r \approx -1.522093720599496 $. I have checked this for all $n<300000$, which obviously is not a large number, but it's striking to see how close some of the roots can get to the boundary of the disk but none pass it.

Answer 1

$T_{393217}(z)$ has a root at $$z\approx-1.50000024999390139088569862099867130326048600135879415656217$$

I have done the computations up to $n=500000$. The only values I found such that the maximum of the modulus of the roots of $T_n$ is greater that $1.5$ are $n=9$ and $n=393217$. I also found (and you probably know already) that the maximum of the modulus of the roots of $T_n$ is strictly greater that $.5$ except for powers of $2$, in which case it is exactly $0.5$.

The following is an histogram of the maximum of the modulus of the roots of $T_n$ for $3\le n\le500000$.