Eldar–Needell–Plan conjecture on the minimal measurement number for low-rank recovery

Let $\mathbb H$ be either $\mathbb R$ or $\mathbb C$, let $\mathcal L_r^{\mathbb H}$ denote the relevant class of matrices of rank at most $r$, and let ${\bf M}_{\mathcal A}$ be the measurement map associated with measurement matrices $\mathcal A=\{A_1,\ldots,A_m\}$. Assume $r\leq n/2$. Eldar–Needell–Plan conjecture. If

$$m<4nr-4r^2,$$

then ${\bf M}_{\mathcal A}$ is not injective on $\mathcal L_r^{\mathbb H}$. The conjecture asks whether the almost-sure sufficient measurement threshold for Gaussian low-rank matrix recovery is also necessary; its status is not determined by the supplied text.