The large-set planar unit-ball decoupling conjecture

Suppose $n=2$, let $Y\subset B_R$ satisfy $|Y|\geq R^{1/2}$, and let $f$ and the pieces $f_\theta$ be as in the unit-ball estimate. Let $\alpha$ denote the exponent appearing in that estimate.

Large-set planar conjecture. If $|Y|\geq R^{1/2}$, then is true with

$$\alpha=\frac{1}{12}.$$

This conjecture is motivated by level-set examples for a quadratic Gauss sum: when $|Y|$ is large, the examples suggest that the exponent $1/12$ remains attainable. The paper presents this as an open proposal in the absence of unforeseen examples.