In this paper, we prove the following statements: (1)~If $X$ is a normal discretely absolutely star-Lindel{ö}f space with $e(X)<\omega_1$, then the Alexandroff duplicate $A(X)$ of $X$ is discretely absolutely star-Lindel{ö}f. (2)~If $X$ is a space with $e(X)\geq\omega_1$, then $A(X)$ is not discretely absolutely star-Lindel{ö}f. The two statements answer a question raised by Song.