Guo–Wang–Zhang's curved Kakeya conjecture for Bourgain's condition

Let $M_0\subset M$ and $\Sigma_0\subset\Sigma$ be closed balls, let $\mathcal V_0=\nabla_\xi\phi(M_0,\Sigma_0)$, and let a $\phi$-Kakeya set be a compact set $K\subset M_0$ containing $\ell_{\xi,v}\cap M_0$ for some $v\in\mathcal V_0$ for every $\xi\in\Sigma_0$. Let Bourgain's condition be the condition defined by

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

Curved Kakeya conjecture. If $\phi$ satisfies Bourgain's condition, then every $\phi$-Kakeya set has Hausdorff dimension $n$.

This is a curved analogue of the classical Kakeya conjecture and is motivated by the connection between Kakeya incidence estimates and Hörmander-type oscillatory integrals. The supplied context gives the implication from the conjectured oscillatory-integral estimate, but does not state a resolution.