Wright's conjecture on phase retrieval by position, Fourier, and unitary measurements

Let $L^2(\mathbb{R})$ be the Hilbert space of square-integrable functions, let $\mathcal{F}$ denote the Fourier transform, and let

$$\mathcal{A}_{QM}=\left\{T:L^2(\mathbb{R})\rightarrow L^2(\mathbb{R},\mu)\mid T\text{ is unitary and }\mu\text{ is a Borel measure}\right\}.$$

A tuple of measurement operators does phase retrieval if its associated measurement data determine a state up to the relevant phase equivalence. Wright's conjecture. There exists a unitary operator $T\in\mathcal{A}_{QM}$ such that $(\operatorname{Id},\mathcal{F},T)$ does phase retrieval. This extends Pauli's phase-retrieval question by asking whether one additional unitary measurement, besides the identity and Fourier transform, suffices. The source attributes the conjecture to R. Wright and states that it remains open.