Masoero–Raimondo string-equation conjecture for the Benjamin–Ono critical profile

Let $U(T,X)$ be the function defined explicitly by the source's formulas for the critical Benjamin–Ono profile, and assume the nonvanishing conjecture $Z\cap\mathbb{R}^2=\emptyset$. For fixed parameter $T$, consider the nonlocal equation

$$X-2UT+4U^3-6U|D_X|U-3|D_X|U^2-4\partial_X^2U=0.$$

Masoero–Raimondo string-equation conjecture. Under the nonvanishing assumption, $U(T,X)$ is a global solution of this equation.

The conjecture connects the explicit critical Benjamin–Ono profile with the nonlocal Painlevé-type equation proposed in the cited formal multiscale analysis. It remains conditional on the nonvanishing conjecture and is not proved in the supplied source.