Classification conjecture for deep holes of extended MDS codes

Let $q$ be a power of a prime, $(q-1)/2le n-k$, and $k<n\le q$. Let ${\bf a}=(\alpha_1,\ldots,\alpha_n)$ and let ${\bf w}$ denote the multiplier vector from the associated MDS code. For ${\bf u}=(u_1,\ldots,u_{n+1})\in {\mathrm{GF}}(q)^{n+1}$, write $\overline{\mathcal{C}_k({\bf a},{\bf 1},\infty)}({\bf u})$ for the extended code obtained by adjoining ${\bf u}$, and let $S_{n+1-k}$ and $T_{\pi,n+1-k}$ be the sets defined in the cited results. Classification conjecture. The extended code $\overline{\mathcal{C}_k({\bf a},{\bf 1},\infty)}({\bf u})$ is MDS if and only if $\rho(\mathcal{C}_{n+1-k}({\bf a},{\bf w},\infty))=k$ and ${\bf u}$ has one of the following forms: (1) ${\bf u}=(\alpha_1f(\alpha_1),\ldots,\alpha_nf(\alpha_n),\delta)$ with $-\delta\notin S_{n+1-k}$; or (2) ${\bf u}=(f_{\pi}(\alpha_1),\ldots,f_{\pi}(\alpha_n),\delta)$ with $\delta\notin T_{\pi,n+1-k}$. Here $f(x)$ and $f_{\pi}(x)$ are the functions specified in Theorem 14(1) and (2), respectively. The conjecture proposes that the two listed families exhaust the possible MDS extensions, equivalently the possible deep-hole forms arising in this setting; its resolution is not supplied in the source.