Conjecture on ergodic real nodal hypersurfaces

Let $(M,g)$ be a real analytic Riemannian manifold with ergodic geodesic flow, and let $\{\varphi_j\}$ be a density-one sequence of ergodic eigenfunctions. For a test function $f$, define the real nodal current by

$$\langle[Z_{\varphi_j}],f\rangle=\int_{Z_{\varphi_j}}f(x)\,d\mathcal H^{m-1},$$

where $d\mathcal H^{m-1}$ is the $(m-1)$-dimensional Hausdorff measure induced by the Riemannian metric, and let $\lambda$ denote the eigenvalue parameter. Ergodic real nodal hypersurface conjecture.

$$\langle[Z_{\varphi_j}],f\rangle\sim\left\{\int_M f\,d\operatorname{Vol}_g\right\}\lambda.$$

This conjecture predicts equidistribution of real nodal hypersurfaces with respect to Riemannian volume on manifolds whose geodesic flow is ergodic. The paper establishes the analogous complex-zero result, while the asserted real nodal asymptotic is left open.