Conjecture on the average number of punctures for GRS covering

Let ${\mathcal C}$ be an $[n,k,d]_q$ generalized Reed–Solomon (GRS) code, and let $\operatorname{GRS\text{-}cover}$ denote the covering algorithm. The average number of punctures is the expected number of coordinates punctured before the algorithm succeeds in returning a codeword within the covering radius.

Puncture-count conjecture. For an $[n,k,d]_q$ GRS code, the average number of punctures needed for $\operatorname{GRS\text{-}cover}$ to succeed in returning a codeword within the covering radius is less than

\begin{cases} c_1(d-1) & \text{when } \operatorname{GRS\text{-}decode \text{ is a unique decoder},\\ c_2 & \text{when } \operatorname{GRS\text{-}decode}=\operatorname{GS} \text{ is the list decoder}, \end{cases}

for some positive constants $c_1<1$ and $c_2$.

The source later states that the first part does not hold in general; the second part remains unresolved there. Thus this combined conjecture is refuted, with its list-decoder component still open.