The stronger second-largest-value conjecture for quadratic exponential sums

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. Stronger form of the conjecture.

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

Moreover, if

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

then

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

This refines the proposed global bound by controlling the next possible magnitude of a quadratic exponential sum. The paper states that the global conjecture has been verified for quadratic $f$ when $n\leq 10$, but the stronger assertion remains open.