Dimension and primality conjecture for CFN sunlet network ideals

Let $J_n$ be the ideal of invariants of the CFN model on the $n$-sunlet network, and let $I_n$ be the ideal generated by all quadratic invariants in $J_n$. Dimension and primality conjecture. For every $n\geq 5$,

$$\dim J_n=2n=\dim I_n$$

and $I_n$ is prime. This conjecture would imply the quadratic generation conjecture, since an ideal of the correct dimension that contains $J_n$ and is prime would force $I_n=J_n$. It is presented as an open conjecture supported by computational evidence.