Nonlinear smoothing conjecture for dispersive equations

Let a nonlinear dispersive equation have dispersion of order $l$ and a nonlinearity of order $k\ge 3$ with a loss of $m$ derivatives. Let $s_0$ be such that, for $s>s_0$, the $H^s$-flow exists and is analytic. Nonlinear smoothing conjecture. The flow exhibits a nonlinear smoothing effect of order

$$\epsilon<\min\{(k-1)(s-s_0),l-m-1\}.$$

This conjecture proposes a general quantitative nonlinear smoothing principle for dispersive equations, based on the order of the dispersion, the order and derivative loss of the nonlinearity, and the regularity threshold for analytic flow. However, the source explicitly states that the conjecture is false when $k=2$; the stated formulation assumes $k\ge 3$.