Cheng–Murray conjecture on deep holes of full-length Reed–Solomon codes

Let $q$ be a prime power, let $\theta$ be a primitive element when relevant, and let ${\rm RS}_k(\mathbb{F}_q)$ denote the full-length Reed–Solomon code of dimension $k$. For a vector in $\mathbb{F}_q^q$, its generating polynomial is the unique polynomial of degree at most $q-1$ agreeing with the vector on all elements of $\mathbb{F}_q$; a deep hole is a vector at covering-radius distance from the code. Cheng–Murray conjecture. For $2 \leq k \leq q-2$, all deep holes of ${\rm RS}_k(\mathbb{F}_q)$ have generating polynomials of degree $k$, except when $q$ is even and $k=q-3$. This conjecture characterizes the deep holes of full-length Reed–Solomon codes, extending the known fact that generating polynomials of degree $k$ produce deep holes. Its status is not resolved by the supplied source.