Non-localized data long-time well-posedness conjecture

Consider a one-dimensional dispersive flow on the real line with a cubic conservative nonlinearity and initial data of size $\epsilon \ll 1$. Non-localized data long-time well-posedness conjecture. The flow has a long-time solution on the $\epsilon^{-8}$ time scale. This focusing-compatible analogue avoids the small-soliton obstruction to global scattering by asserting only a long-time lifespan; the source presents it as a conjectural optimal bound.