Linear recurrence conjecture for the A_{1,N-1} T_z-systems

Let $N\geq 3$. Consider the iterates $x_n$ of the $\tilde{A}_{1,N-1}$ T$_z$-system

$$x_{n+N}x_n={\cal Z}_n(x_{n+N-1}x_{n+1}+1),\qquad {\cal Z}_{n+N-2}{\cal Z}_n=1,$$

associated with the Y-system

$$y_{n+N}y_n=(1+y_{n+N-1})(1+y_{n+1}).$$

Here $s=(N-1)(N-2)$, and a first integral is a quantity constant along the iterates.

Linear recurrence conjecture. The iterates satisfy a constant-coefficient linear recurrence relation of order $4(N-1)(N-2)$. For odd $N$, it has the form

$$x_{n+4s}-\mathrm{A}x_{n+3s}+\mathrm{B}x_{n+2s}-\mathrm{A}x_{n+s}+x_n=0,$$

while for even $N$ it has the form

$$x_{n+4s}-\mathrm{C}x_{n+2s}+x_n=0,$$

where $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ are first integrals. The conjecture would extend the explicitly verified linear relation for the $T_z$-system at $N=4$ to the whole affine Dynkin family; the source reports extensive numerical evidence, but does not establish the relations in general.