Asymptotic Hamming bound for unrestricted data-syndrome codes

Let $d$ be the distance parameter, let $n$ denote the block length, and let unrestricted DS codes be quantum data-syndrome codes without a non-degeneracy restriction. Write the Hamming bound for non-degenerate DS codes as

. **Asymptotic Hamming-bound conjecture.** For any $d$, there \exists $n(d)$ such that, for all $n\geq n(d)$, the Hamming bound

holds for unrestricted DS codes. The conjecture is motivated by numerical observations that the Hamming bounds for unrestricted and non-degenerate DS codes coincide for sufficiently large $n$; the supplied text gives no proof or resolution.