Quantum complexity in the intermediate-precision regime

Determine the quantum query complexity of convex optimization in the intermediate precision regime between the regimes covered by $epsilon=\Omega(1/\sqrt{n})$ and $epsilon=O(1/n^2)$.

Background

The paper combines its high-accuracy lower bounds with earlier accuracy-dependent results to characterize convex-optimization query complexity up to logarithmic factors when the target accuracy is either at least on the order of $1/\sqrt n$ or at most on the order of $1/n^2$.

The authors explicitly state that the complexity for the intervening range of target accuracies is unresolved. This question concerns the missing dimension-accuracy tradeoff between the two established regimes.