The quadratic exponential-sum bound and its extremal polynomials

Let $m\geq 3$ be odd, let $n$ be a non-negative integer, and let $f$ be a quadratic polynomial in $n$ variables modulo $m$. Write $q=2\cos\left(\frac{\pi}{2m}\right)$ and let $S(f,n,m)$ denote the associated normalized quadratic exponential sum. The quadratic exponential-sum conjecture.

$$\left|S(f,n,m)\right|\leq\left(\frac{q}{2}\right)^{\left\lfloor\frac{n+1}{2}\right\rfloor}.$$

Moreover, the upper bound is attained by all polynomials of the form

$$c(\pm x_1x_2\pm x_3x_4\pm\cdots\pm x_{n-1}x_n)$$

when $n$ is even, and by any polynomial of the form

$$c(\pm x_1x_2\pm x_3x_4\pm\cdots\pm x_{n-1}x_n\pm x_{n+1})$$

when $n$ is odd, where $c=\left\lfloor\frac{m+1}{4}\right\rfloor$. The conjecture would give a sharp, exponentially decreasing bound for incomplete quadratic exponential sums; it was verified in the paper for quadratic polynomials when $n\leq 10$, while the general case remains open.