Cancellation conjecture for quadratic exponential sums

Let $q(l)$ be a quadratic polynomial, with the sum taken over integers $l$ satisfying $q(l)<x$, and let $x$ tend to infinity. Cancellation conjecture.

$$\sum_{q(l)<x}(-1)^l e^{\sqrt{x-q(l)}}=e^{o(\sqrt{x})}.$$

This conjectures cancellation substantially stronger than a probabilistic heuristic would suggest for a broad class of quadratic sums. The surrounding results prove related estimates for quadratic polynomials under additional hypotheses, but the displayed general assertion is not established here.