The Chowla conjecture for Kloosterman sums

Fix an integer $a\ne0$, positive integers $\nu_1,\ldots,\nu_s$, and integers $h_s>\cdots>h_1\geqslant0$. Let ${\mathcal K}_m^*(a)$ be the normalized Kloosterman-sum quantity and let $\overline{\mathsf H}_{a}^*(M)$ be the associated mean-square scale. Chowla conjecture for Kloosterman sums. As $M\to\infty$, if $\nu_1,\ldots,\nu_s$ are not all even, then

$$\left|\frac{1}{M}\sum_{m=1}^{M}{\mathcal K}_{m+h_1}^*(a)^{\nu_1}\cdots{\mathcal K}_{m+h_s}^*(a)^{\nu_s}\right|=o\left(\frac{\overline{\mathsf H}_{a}^*(M)}{M}\right)^{\nu_1+\cdots+\nu_s};$$

whereas if $\nu_1,\ldots,\nu_s$ are all even, then

$$\left|\frac{1}{M}\sum_{m=1}^{M}{\mathcal K}_{m+h_1}^*(a)^{\nu_1}\cdots{\mathcal K}_{m+h_s}^*(a)^{\nu_s}\right|\leqslant A\left(\frac{\overline{\mathsf H}_{a}^*(M)}{M}\right)^{\nu_1+\cdots+\nu_s},$$

for some constant $A\geqslant1$ depending only on $a$ and uniform in all other parameters. This is presented as a Chowla-type assertion for Kloosterman sums, motivated by the expected randomness of these sums; the source states that stronger explicit decay estimates are needed because the exact order of $\overline{\mathsf H}_{1}^*(M)$ is not known.