Almost-everywhere uniqueness conjecture for complex phase retrieval least squares

Let $A\in \mathbb{C}^{m\times d}$ have the phase retrieval property in $\mathbb{C}^d$, meaning that $A$ determines every vector in $\mathbb{C}^d$ up to the relevant phase ambiguity from its measurement magnitudes. For ${\bm b}\in\mathbb{R}^m$, consider the nonlinear least-squares program

$$\min_{{\bm x}\in\mathbb{C}^d}\|\lvert A{\bm x}\rvert-{\bm b}\|_2^2.$$

Almost-everywhere uniqueness conjecture. For all vectors ${\bm b}\in\mathbb{R}^m$ except for a measure zero set, this program has a unique solution. The real analogue is established by the preceding theorem; the conjectured complex analogue concerns uniqueness for almost every real data vector.