Generic sharpness conjecture for local well-posedness thresholds

Let $s$ denote the Sobolev regularity exponent, and let Theorems~ and be the local well-posedness results stated in the source for the two-dimensional and $n$-dimensional cubic problems. Generic sharpness conjecture for local well-posedness thresholds. The range of $s$ in those theorems is generically sharp, except possibly at the endpoint. The source attributes the expected obstruction in higher dimensions to trapping and notes that this differs from the apparent single-wave-packet obstruction in one dimension; the conjecture remains open.