KöMaL 2014 B4645

Let $H_{1}=\{1, 3, 5, \ldots, 2n-1\}$ and $H_{2}=\{1+k, 3+k, 5+k, \ldots, 2n-1+k\}$, where $n$ and $k$ are any positive integers. Is there an appropriate $k$ for every $n$ such that the product of all elements of the set $H_{1}\cup H_{2}$ is a perfect square?