Completeness conjecture for deep holes of twisted Reed–Solomon codes

Let $q$ be a prime power, let $\theta$ specify the twist defining the twisted Reed–Solomon code ${\rm TRS}_k(\mathbb{F}_q,\theta)$, and let $\mathcal{S}_{k,\theta}$ be the source-defined set of admissible correction polynomials. A deep hole is a word at covering-radius distance from the code. Completeness conjecture for twisted Reed–Solomon codes. For $2 \leq k \leq q-4$, all deep holes of ${\rm TRS}_k(\mathbb{F}_q,\theta)$ have generating polynomials

$$f(x)=ax^{k}+f_{k,\theta}(x)$$

with $a \neq 0$ and $f_{k,\theta}(x) \in \mathcal{S}_{k,\theta}$. This conjecture proposes that the described family gives all deep holes for the indicated parameter range. The paper presents the problem as a preliminary exploration and notes that the relatively small-dimension case remains an interesting direction for future work; no resolution is supplied.