Random-initialization success conjecture for alternating projections in phase retrieval

Let $x_0\in\mathbb{C}^n$ be the target vector, let $A\in\mathbb{C}^{m\times n}$ be the sensing matrix, and let alternating projections reconstruct $x_0$ from the phaseless measurements $|Ax_0|$. An initialization is isotropic when its law is invariant under linear unitary transformations. Random-initialization success conjecture. For every fixed $\epsilon>0$, there is a sufficiently large constant $C>0$ such that, whenever $m\geq Cn$, alternating projections started from a random isotropic initialization converge to the true solution with probability at least $1-\epsilon$. The conjecture predicts that, although stagnation points remain in the linear-measurement regime, their attraction basins are small enough that random initialization succeeds with probability arbitrarily close to one when the measurement-to-dimension ratio is sufficiently large.