Capacity-matrix probabilistic recovery bound

Let $\boldsymbol{Q}$ be the capacity matrix, let $\boldsymbol{Q}^{V}$ be the vector of values $Q_{i,j}$ with $i<j$ in $\Omega$, and let $E_Q$ and $\sigma_Q^2$ denote its mean and variance, respectively:

$$\sigma_Q^2=\frac{2}{L(L-1)}\sum_{i<j\in\Omega}(Q_{i,j}-E_Q)^2.$$

For an even integer $ll$ and a support $\Gamma$ sampled uniformly at random from $\Omega$, write $P(\ell)$ for the probability that $\Gamma$ admits $\ell_1$-recovery. Capacity-matrix recovery conjecture. For any $1\leq \ell<\frac{1}{E_Q}$, a support $\Gamma$ of even size $\ell$ admits $\ell_1$-recovery with probability

$$P(\ell)>\frac{\left(\frac{1}{2}-\frac{\ell}{2}E_Q\right)^2}{\frac{\ell}{2}\sigma_Q^2+\left(\frac{1}{2}-\frac{\ell}{2}E_Q\right)^2}.$$

The claim is conjectural because it relies on a property used without proof; the paper indicates that further discussion appears in Appendix B. It provides a probabilistic recovery guarantee based on the mean and variance of the capacity-matrix entries.