The entropy-based measurement conjecture for binary sparse recovery

Let $A$ be the measurement matrix in the linear system

$$Ax=b.$$

Assume that the entries of $A$ are independent and identically distributed according to an absolutely continuous distribution. Let $x$ be a binary $k$-sparse solution, and let $H$ denote the binary entropy function.

Entropy-based measurement conjecture. For sufficiently large $n$, the linear program referred to as LP~ exactly recovers $x$ from $nH(k/n)/2$ measurements.

This conjecture proposes an information-theoretic measurement threshold for exact recovery of binary sparse solutions under absolutely continuous measurement ensembles. The supplied text gives no resolution, so its status remains open.