Sharpness conjecture for the local well-posedness regularity range

Consider the cubic problems denoted by

$$and$$

. Sharpness conjecture. Generically, the problem

is ill-posed in $H^s$ for $s<1$, while

is ill-posed in $H^s$ for $s<2$. The theorem establishes local well-posedness above these thresholds, while the endpoint cases $s=1$ for

and $s=2$ for

remain open.