The conjecture on uncorrelated classical and modular-symbol Kloosterman sums

For $m,n\neq0$, define

$$\rho_{m,n}(x)=\frac{\#\{N\mid c\leq x:\operatorname{sign}S(m,n;c)=\operatorname{sign}S^*(m,n;c)\}}{\#\{N\mid c\leq x\}},$$

and let $r(S,S^*)(x)$ be the Pearson correlation of the sequences $S(m,n;c)$ and $S^*(m,n;c)$ over the dyadic window $W_x=\{N\mid c:x<c\leq2x\}$. Uncorrelation conjecture. For any $m,n\neq0$,

$$\rho_{m,n}(x)\to\frac12\quad\text{and}\quad r(S,S^*)(x)\to0 \qquad (x\to\infty).$$

The conjecture formalizes the numerical evidence that the signs and values of the two types of Kloosterman sums are asymptotically uncorrelated; the supplied source gives numerical support but no proof.