Exponential correlation decay for non-classical polynomial phases

Let $p$ be a prime and let $k\geq p+2$ be an integer. For each $n$, consider functions $f\colon\mathbb F_p^n\to\mathbb C$ with $\|f\|_{\infty}\leq 1$ and Gowers norm $\|f\|_{U^k}\geq c_{p,k}>0$, where $c_{p,k}>0$ depends only on $p$ and $k$. Let $P\in\operatorname{Poly}_{\leqslant k-1}(\mathbb F_p^n\to\mathbb F_p)$ range over classical polynomials of degree at most $k-1$. Correlation-decay conjecture. For every $n$ there exists such an $f$ satisfying, for every such polynomial $P$,

$$\left|\mathbb E_{x\in\mathbb F_p^n} f(x)e_p(-P(x))\right|\leq \exp(-\Omega_{p,k}(n)).$$

This would improve the iterated-logarithmic correlation bound obtained by the paper's Ramsey-theoretic argument. It asserts that functions with positive $U^k$-norm can have exponentially small correlation with every classical polynomial phase of degree at most $k-1$, in the range where non-classical polynomials are necessary for the inverse theorem.