Random puncturing conjecture for list recovery of Reed–Solomon codes

Let $d$ be fixed, and consider the degree-$d$ Reed–Solomon code with evaluation set $FbbF_q$. A random puncturing is obtained by retaining a random subset of evaluation coordinates, and the resulting code has its usual rate and list-recovery parameters. Random puncturing conjecture. The degree-$d$ Reed–Solomon code with evaluation set $FbbF_q$ can be randomly punctured to rate

$$\Omega_d\left(\frac{1}{\log q}\right)$$

so that it is $(\Omega(q),1/2)$-list recoverable with high probability. This conjecture proposes that the logarithmic-in-$q$ rate is essentially the best possible rate for random puncturing while retaining list recoverability with list size proportional to $q$ and agreement parameter $1/2$.