Phase retrieval conjecture for the harmonic oscillator Schrödinger equation

Let $u_0,v_0\in L^2(\mathbb{R}^d)$, and let $u$ and $v$ solve the harmonic-oscillator Schrödinger equations

$$\left\{\begin{matrix}i\partial_tu(t,x)=-\Delta u(t,x)+|x|^2u(t,x)\\ u(0,x)=u_0\end{matrix}\right.$$

and

$$\left\{\begin{matrix}i\partial_tv(t,x)=-\Delta v(t,x)+|x|^2v(t,x)\\ v(0,x)=v_0\end{matrix}\right..$$

Harmonic-oscillator phase-retrieval conjecture. If $|u(t,x)|=|v(t,x)|$ for every $t\in\mathbb{R}$ and every $x\in\mathbb{R}^d$, then there exists $c\in\mathbb{R}$ such that $u_0=cv_0$. The conjecture concerns recovery of initial data from the modulus of the solution at all times and positions; the source presents it as a natural conjecture motivated by the semiclassical Wigner and circular Radon-transform picture, and the supplied material does not establish its resolution.