ZWK conjecture on the covering radius of projective Reed–Solomon codes

Let $q$ be a prime power and let $\operatorname{PRS}(k)$ denote the projective Reed–Solomon code of dimension $k$ and length $q+1$. For $2\le k\le q-2$, let $\rho(\operatorname{PRS}(k))$ be its covering radius. ZWK conjecture. The covering radius is

$$\rho(\operatorname{PRS}(k))=\begin{cases}q-k+1,&\text{if }q\text{ is even and }k\in\{2,q-2\},\q-k,&\text{otherwise}. \end{cases}$$

The paper states that the covering radius is known for the boundary dimensions $k\in\{1,q,q+1\}$ and for $k=q-1$, while the range $2\le k\le q-2$ is open there; the conjecture is attributed to ZWK.