The $4M-4$ phase-transition conjecture for complex phase retrieval

Let $\Phi$ be the intensity-measurement matrix for signals in $\mathbb{C}^M$, with $N$ measurements, and write $\mathrm{Inj}[\Phi;\mathbb{C}^{M\times N}]$ for the property that the measurements are injective. The $4M-4$ conjecture. $\mathrm{Inj}[\Phi;\mathbb{C}^{M\times N}]$ exhibits a phase transition at $N=4M-4$. This conjecture concerns the algebraic-geometric threshold suggested by the variety of matrices of rank at most $2$; the source records that the relevant part is proved by exhibiting a nonempty complement, so the conjecture is resolved.