Sarnak's unbounded-component conjecture for monochromatic random waves

Let $d\geq 3$, and let $f$ be the monochromatic random wave in $\mathbb{R}^d$, meaning the centered stationary smooth Gaussian field whose covariance is the Fourier transform of the uniform measure on the $(d-1)$-dimensional sphere. For a level $\ell\in\mathbb{R}$, consider the level set $\{f=\ell\}$ and its intersection with the embedded copy of $\mathbb{R}^3$.

Sarnak's conjecture. There exists $\delta>0$ such that, for every $\ell\in[-\delta,\delta]$, almost surely $\{f=\ell\}\cap\mathbb{R}^3$ contains an unbounded component.

The conjecture extends the paper's unbounded-component results beyond positively correlated fields: the monochromatic random wave has oscillatory covariance with polynomial decay. It was stated in the cited work of Sarnak and remains open.