  For $k \ge 2$ and a positive integer $d_0$, we show that if there exists no
quaternary Hermitian linear complementary dual $[n,k,d]$ code with $d \ge d_0$
and Hermitian dual distance greater than or equal to $2$, then there exists no
quaternary Hermitian linear complementary dual $[n,k,d]$ code with $d \ge d_0$
and Hermitian dual distance $1$. As a consequence, we generalize a result by
Araya, Harada and Saito on the nonexistence of some quaternary Hermitian linear
complementary dual codes.
