Cynthia Vinzant just disproved part (a) of the conjecture, see here!
We posed the conjecture two years ago, and there was even a monetary motivation for it! I will not go into details regarding the conjecture, you can read more about it either on a previous blog post of mine or a post of Dustin Mixon. In a nutshell it tries to predict how many phaseless measurements are needed to identify a complex vector of dimensions. The conjecture was that (a) measurements are always needed and (b) measurements suffice (as long as sufficiently generic).
Part (b) had been solved (positively) around a year ago Aldo Conca, Dan Edidin, Milena Hering, and Cynthia Vinzant (see the paper here) and part (a) was just disproved by Cynthia with an example in four dimensions.