True or false ? Sqares #2

For all positive integers $n$ , there exists a positive integer $k$ , such that $\underbrace{\overline {nn...n} }_\text{k times}$ is a perfect sqaure.

True or false ?

Note : $\overline {nn}$ stands for the concatenation of $n$ and $n$ in their decimal representation. Examples : For $n=5$ , $\overline {nn}=55$ and for $n=1663$ , $\overline {nnn}=166316631663$ .