Consider the following *somos-like* sequence
$$x_n=\frac{x_{n-1}^2+x_{n-2}^2}{x_{n-3}}.$$
It's known that $x_n$ is a Laurent polynomial in $x_0, x_1$ and $x_2$. I got interested in the denominators of the sequence $x_n$. Some initial observations indicate particular structures regarding the exponents of the denominators, so I am tempted to ask:

Question:Is this true? $$x_n=P_n(x_0,x_1,x_2)x_0^{1-F_{n-3}}x_1^{1-F_{n-4}}x_2^{1-F_{n-5}},$$ where $P_n$ is some polynomial and $F_n$ is the Fibonacci sequence.