On recognition of direct powers of finite simple linear groups by spectrum

The spectrum of a finite group is the set of its element orders. We give an affirmative answer to Problem 20.58(a) from the Kourovka Notebook proving that for every positive integer $k$, the $k$-th direct power of the simple linear group $L_n(2)$ is uniquely determined by its spectrum in the class of finite groups provided $n$ is a power of $2$ greater than or equal to $56k^2$.