  In 1968, Golomb and Welch conjectured that there is no perfect Lee codes with
radius $r\ge2$ and dimension $n\ge3$. A diameter perfect code is a natural
generalization of the perfect code. In 2011, Etzion (IEEE Trans. Inform.
Theory, 57(11): 7473--7481, 2011) proposed the following problem: Are there
diameter perfect Lee (DPL, for short) codes with diameter greater than four
besides the $DPL(3,6)$ code? Later, Horak and AlBdaiwi (IEEE Trans. Inform.
Theory, 58(8): 5490--5499, 2012) conjectured that there are no $DPL(n,d)$ codes
for dimension $n\ge3$ and diameter $d>4$ except for $(n,d)=(3,6)$. In this
paper, we give a counterexample to this conjecture. Moreover, we prove that for
$n\ge3$, there is a linear $DPL(n,6)$ code if and only if $n=3,11$.
