Do primitive divisor theorems apply to the numerator sequence of this rational quadratic orbit?

I started from a modular experiment for integers of the form

$$ N=2k+11, $$

where I iterated

$$ U_0=k+1,\qquad U_{n+1}=U_n^2-k\pmod N, $$

and called an integer $N$ captured if the orbit reached a solution of

$$ x^2\equiv k\pmod N. $$

After rewriting the iteration symbolically using

$$ k=\frac{N-11}{2}, $$

I found that every iterate can be written as

$$ U_n=\frac{P_n(N)}{2^{2^n}}, $$

with $P_n(N)\in\mathbb Z[N]$, and that

$$ N\text{ is captured at iteration }n \iff N\mid P_{n+1}(0). $$

Evaluating at $N=0$ recovers the rational orbit

$$ U_0=-\frac92,\qquad U_{n+1}=U_n^2+\frac{11}{2}, $$

so the captured primes are precisely the prime divisors of the numerator sequence $P_n(0)$ of this single rational orbit.

Edit.

Following the suggestion in the comments, here is the sequence more explicitly.

The iteration is

$$ U_0=-\frac92,\qquad U_{n+1}=U_n^2+\frac{11}{2}. $$

Writing each iterate in lowest terms,

$$ U_n=\frac{a_n}{b_n}, $$

I define

$$ P_n(0)=|a_n|. $$

The first terms are

$$ P_1(0)=103, $$

$$ P_2(0)=10697=19\cdot563, $$

$$ P_3(0)=114427217=173\cdot677\cdot977, $$

$$ P_4(0)=13093639452262846897, $$

with

$$ 61\mid P_4(0). $$

This is the sequence referred to in the question.

My question is: since the modular experiment ultimately reduces to the numerator sequence of one rational quadratic orbit, do the existing primitive divisor theorems in arithmetic dynamics imply that, apart from finitely many exceptions, every new term $P_n(0)$ introduces a primitive prime divisor?