From Our short paper

For polynomial $F$ with integer coefficients, define the recurrence $f(n)=F(n,f(n-1),f(n-2),...,f(n-d))$. We conjecture that $f(n)$ satisfy Somos like sequence
$f(n)=\frac{G(f(n-1),...,f(n-d_1))}{H(f(n-1),...,f(n-d_2))}$ for polynomials $G$,$H$ which do not depend on $n$.

The strong conjecture is for all $F$ and the weak conjecture is about $F$ which is linear in $f(n-i)$.

It solves special parametric cases like $f(n)=F(n,f(n-1),f(n-2))=(f(n-1)^2)(b_1 n+b_2)+f(n-2)(b_3 n+b_4)$ for all integers $b_i$ and $f(n)=f(n-1)(n^2+n+2)+f(n-2)(3 n+ 5 )$. The proofs are based on finding algebraic dependency of polynomials.

Q1 Are the conjectures true?

Added later

In comments was suggested existence of algebraic dependency not depending on $n$ using resultants of consecutive terms.

This doesn't prove the conjectures, since additional linearity "luck" is needed.