The Hermite Wigner zero-set rigidity conjecture

Let $Wf(z)$ be the Wigner distribution of some function $f\in L^2(\mathbb{R})$, with bounded nodal set $\mathcal{N}(Wf)$, and suppose that $Wf$ is centered at $z=0$. For a Hermite function $h_k$, write $\mathcal{N}(Wh_k)$ for the nodal set of its Wigner distribution. Hermite Wigner zero-set rigidity conjecture. If

$$\mathcal{N}(Wh_k)\subset \mathcal{N}(Wf),$$

then, and only then, $Wf(z)=Wh_k(z)$ everywhere. This conjecture extends the verified cases for the first, second, and third Hermite functions to all Hermite functions; the paper states that it is proved conditional on a conjecture concerning divisibility of Laguerre polynomials.