Hamilton-Jacobi conjecture for logarithmic Husimi limits

Let $M_\epsilon$ be the complexified manifold under consideration, let $\varphi_{j}^{\mathbb{C}}$ be complexified eigenfunctions with parameters $\lambda_j$, and suppose that along a subsequence

$$\frac{1}{\lambda_{j_k}}\log\left|\varphi_{j_k}^{\mathbb{C}}(z)\right|^2\to G(z).$$

Hamilton-Jacobi conjecture. Any such limit $G$ solves the Hamilton-Jacobi equation

$$(\nabla_{\mathbb{C}}G)^2=1.$$

The claim describes a proposed equation for limits of normalized logarithms of complexified eigenfunctions. The source presents it as a heuristic principle and gives no resolution status.