Complexified-geodesic nodal-zero equidistribution conjecture

Let $(M^2,g)$ be a real analytic Riemannian surface with ergodic geodesic flow, and let $\gamma_{x,\xi}$ be a geodesic satisfying the QER hypothesis. Let $S_\epsilon=\{t+i\tau\in\mathbb C:|\tau|\leq\epsilon\}$ be a complex strip on which the geodesic and eigenfunctions analytically continue. Complexified-geodesic equidistribution conjecture. There exists a density-one subsequence of eigenvalues $\lambda_{j_k}$ such that, for every $f\in C_c(S_\epsilon)$,

$$\lim_{k\to\infty}\sum_{(t+i\tau):\,\varphi_{j}^{\mathbb C}(\gamma_{x,\xi}^{\mathbb C}(t+i\tau))=0}f(t+i\tau)=\int_{\mathbb R}f(t)\,dt.$$

Thus the complex zeros on the complexified geodesic condense on the real geodesic and become uniformly distributed with respect to arc length. The source says this appears to have been proved, but conservatively states it as a conjecture; the database status therefore remains open.