Guo–Wang–Zhang's Hörmander dichotomy conjecture

Let $M$ and $\Sigma$ be the parameter spaces for a Hörmander-type phase function $\phi$, and let $T_\phi^\lambda$ be the associated oscillatory integral operator. Bourgain's condition means that there is a smooth function $\lambda:M\times\Sigma\to\mathbb{R}$ such that

$$(G(\mathbf{x},\xi)\cdot\nabla_\mathbf{x})^2\nabla_\xi^2\phi(\mathbf{x},\xi)=\lambda(\mathbf{x},\xi)(G(\mathbf{x},\xi)\cdot\nabla_\mathbf{x})\nabla_\xi^2\phi(\mathbf{x},\xi).$$

Hörmander dichotomy conjecture. The operators $T_\phi^\lambda$ satisfy the operator boundedness estimate for $p>\frac{2n}{n-1}$ if and only if $\phi$ satisfies Bourgain's condition.

This conjecture is intended to answer the best-case Hörmander oscillatory-integral question and unify the Bochner–Riesz and Fourier restriction conjectures. The supplied context records counterexamples below the stated exponent when Bourgain's condition fails, but gives no resolution of the equivalence.