Almost perfect linear Lee codes of packing radius 2 only exist for small dimensions

It is conjectured by Golomb and Welch around half a century ago that there is no perfect Lee codes $C$ of packing radius $r$ in ${\mathbb{Z}}^n$ for $r\ge 2$ and $n\ge 3$. Recently, Leung and the second author proved this conjecture for linear Lee codes with $r=2$. A natural question is whether it is possible to classify the second best, i.e., almost perfect linear Lee codes of packing radius $2$. We show that if such codes exist in ${\mathbb{Z}}^n$, then $n$ must be $1,2,11,29,47,56,67,79,104,121,134$ or $191$.