Patterson's asymptotic conjecture for cubic exponential sums

Let $p$ run through the primes with $p\equiv 1\pmod 3$, and define

$$S_p=\sum_{a=1}^{p}e\left(\frac{a^3}{p}\right),$$

where $e(y)=e^{2\pi i y}$. Let

$$d=\frac{2(2\pi)^{2/3}}{5\Gamma(\frac{2}{3})},$$

where $\Gamma$ is the classical gamma function. Patterson's conjecture. As $X\to\infty$,

$$\sum_{\substack{p\leq X\\ p\equiv 1\bmod 3}}\frac{S_p}{2\sqrt{p}}\sim d\frac{X^{5/6}}{\log X}.$$

The conjecture concerns the first moment of normalized cubic exponential sums and their distribution over primes congruent to $1$ modulo $3$. The preceding Kummer frequency conjecture was disproved by Heath-Brown and Patterson; the source gives no resolution status for Patterson's conjecture itself.