Robust nullspace property conjecture for sparse Bernoulli matrices

Let $A\in\{0,1\}^{m\times n}$ be a matrix whose entries are independent and identically distributed Bernoulli $p$ random variables. The robust nullspace property conjecture. There exist constants $C_1,C_2>0$ such that, if

$$m\geq\max\left\{C_1s\log\frac{en}{s},C_2\frac{\log n}{p}\right\},$$

then $A$ satisfies the $\ell_1$ robust nullspace property of order $s$ for some $\rho<1$ and $\tau>0$. Moreover, any constant $C_2>1$ can be taken. This conjecture predicts the sharp measurement threshold and a phase transition for robust sparse recovery with sparse Bernoulli matrices; the supplied text does not give a resolution.