Conjecture on polynomial exponential sums becoming kloostermanian functions

For $u,v\in\mathbb{Z}$, $c\in\mathbb{N}$, and polynomials $g,h\in\mathbb{Z}[x]$, define

$$A(ug,vh;c)=\sum_{x \, (\operatorname{mod}{ c})}e\left(\frac{ug(x)+vh(x)}{c}\right),$$

and

$$\widetilde{A}(ug,vh;c)=\sum_{x \, (\operatorname{mod}{ c})}\left(\frac{x}{c}\right)e\left(\frac{ug(x)+vh(x)}{c}\right).$$

Also define the corresponding sums over invertible residue classes by

$$A^{\times}(ug,vh;c)=\sum_{\substack{x \, (\operatorname{mod}{ c})\\\x\overline{x}\equiv1\\,(\operatorname{mod}{ c})}}e\left(\frac{ug(x)+vh(\overline{x})}{c}\right)$$

and

$$\widetilde{A}^{\times}(ug,vh;c)=\sum_{\substack{x \, (\operatorname{mod}{ c})\\\x\overline{x}\equiv1\\,(\operatorname{mod}{ c})}}\left(\frac{x}{c}\right)e\left(\frac{ug(x)+vh(\overline{x})}{c}\right).$$

Here the four sums are T-multiplicative arithmetic functions of $c$.

Polynomial exponential-sum conjecture. If $g$ and $h$ satisfy

$$\min\\{\deg(g),\deg(h)\\}\geqslant1,\qquad \max\\{\deg(g),\deg(h)\\}\geqslant3,$$

then there exists an integer pair $a,b$ such that $A(ag,bh;c)$ is a kloostermanian arithmetic function and $\widetilde{A}(ag,bh;c)$ is an almost-kloostermanian arithmetic function. If $g$ and $h$ satisfy $\min\\{\deg(g),\deg(h)\\}\geqslant1$, then there exists an integer pair $a,b$ such that $A^{\times}(ag,bh;c)$ is a kloostermanian arithmetic function and $\widetilde{A}^{\times}(ag,bh;c)$ is an almost-kloostermanian arithmetic function.

These assertions propose that suitable integer rescalings of polynomial exponential sums, including their quadratic-character twists and invertible-variable variants, can exhibit the same arithmetic behavior as Kloosterman-type functions. The supplied text gives no resolution, so the conjecture remains open.