Binary-extension QMDS existence conjecture

For $q=2^m$, consider quantum MDS codes over systems of dimension $q$, with parameters written as $[[n,n-6,4]]_{2^m}$. Binary-extension QMDS existence conjecture. If $q=2^m$ and $q\ne4$, then there exist QMDS codes with parameters

$$[[n,n-6,4]]_{2^m}$$

for every integer $n$ satisfying $6\le n\le4^m+2=q^2+2$. Computational results establish the asserted range only for the values examined in the paper, so the general claim remains open.