Quartic lifespan conjecture for small-data ILW solutions

The Intermediate Long Wave equation has small initial data of size $\epsilon$ and dispersion relation

$$\tau=xi^2 coth xi-xi,$$

which is KdV-like at low frequencies and Benjamin–Ono-like at high frequencies. Quartic lifespan conjecture. The result of the quartic-time theorem should hold up to a quartic time

$$t lesssim \epsilon^{-3}.$$

This expectation is motivated by the quartic lifespan known for small-data KdV solutions and by the emergence time of solitary waves; the supplied text does not establish whether the conjecture is resolved.