Vinzant's refined conjecture on random complex phase retrieval

Let $N=4M-5$ and let $A\in\mathbb C^{N\times M}$ have entries drawn independently from the standard complex Gaussian distribution, where a standard complex Gaussian has the law of $a+ib$ with independent $a,b\sim\mathcal N(0,\frac12)$. Let $p_M$ be the probability that the phase retrieval map $\Phi_A$ is injective. Vinzant's refined conjecture. One has

$$p_M<1\quad\text{for all }M,$$

and

$$\lim_{M\to\infty}p_M=0.$$

The paper proves the first part by showing that the non-injectivity probability is positive for every $M$, while the asymptotic assertion remains open in the supplied text.