Conjecture on puncture complexity and covering radius for fixed-rate GRS codes

Let ${\mathcal C}$ be an $[n,k,d]_q$ generalized Reed–Solomon (GRS) code of fixed rate $R$, let $\operatorname{GRS\text{-}cover}$ be the covering algorithm, let $\operatorname{GS}$ be the Guruswami–Sudan list decoder, and write $\rho_{\mathrm H}({\mathcal C})$ for the Hamming covering radius of ${\mathcal C}$. The notation $\Theta(\rho_{\mathrm H}({\mathcal C}))$ means that the average number of punctures is bounded above and below by positive constant multiples of this covering radius.

Fixed-rate puncture conjecture. For an $[n,k,d]_q$ GRS code ${\mathcal C}$ of fixed rate $R$, the average number of punctures needed for $\operatorname{GRS\text{-}cover}$ to succeed with $\operatorname{GRS\text{-}decode}=\operatorname{GS}$ is

$$\Theta\bigl(\rho_{\mathrm H}({\mathcal C})\bigr).$$

The source presents this as a proposed modification of the list-decoder part of the earlier conjecture. It is not resolved in the supplied text.