Pierce's discrete quadratic Carleson conjecture on $\ell^2(\mathbb Z)$

Let $f\in\ell^2(\mathbb Z)$ and write $e(t)=e^{2\pi i t}$. Pierce's discrete quadratic Carleson conjecture. The inequality

$$\Bigl\lVert \sup_{0 \leq \lambda \leq 1} \Bigl\lvert \sum_{n\neq 0} f(x-n)\frac{e(\lambda n^2)}{n}\Bigr\rvert\,\Bigr\rVert_{\ell^2(\mathbb Z)} \lesssim \lVert f\rVert_{\ell^2(\mathbb Z)}$$

holds. This is a discrete analogue of Stein's maximal quadratic operator and is intended to connect oscillatory integral estimates with Bourgain's discrete polynomial ergodic theory. The supplied source attributes the conjecture to Lillian Pierce and gives no evidence of a resolution.