Showing a certain dense subset of $\mathcal{C}^1([0,1]^2,\mathbb{R}^4)$ is residual.

Question

For every $f\in X\doteq \mathcal{C}^1([0,1]^2,\mathbb{R}^4)$, define $S_f=\{x\in[0,1]^2:f_1(x)f_2(x)-f_3(x)f_4(x)=0\}$. I am wondering if the set $G=\{f\in X:S_f\text{ is contained in the union of finitely many curves }C_f^1,...,C_f^{n_f}\}$ is residual.

Here are some observations about $G$:

1. I showed that $G$ is dense, $G$ is not open, and $G$ has a countable dense subset.
2. If I replace $C_f^1,...,C_f^{n_f}$ with fixed curves $C^1$,...,$C^n$ then $G$ is residual.
3. If I replace $X$ with $\mathcal{C}^1([0,1],\mathbb{R})$ make $C_f^{n_1}$,...,$C_f^{n_f}$ points instead of curves, then $G$ is residual.