If $\beta^*=\mathrm{arg\,min}_{\beta} \|y-X\beta\|^2_2+\lambda\|\beta\|_1$, can $\|\beta^*\|_2$ increase when $\lambda$ increases?

I think this is possible. Although $\|\beta^*\|_1$ does not increase when $\lambda$ increases (my [proof](http://math.stackexchange.com/a/1223116/11014)), $\|\beta^*\|_2$ can increase. The figure below shows a possibility. When $\lambda$ increases, if $\beta^*$ travel (linearly) from $P$ to $Q$, then $\|\beta^*\|_2$ increases while $\|\beta^*\|_1$ decreases. But I don't know how to construct a concrete example (i.e., construct $X$ and $y$), so that the profile of $\beta^*$ demonstrate this behavior. Any ideas? Thank you.

![enter image description here][1]


  [1]: https://i.sstatic.net/LP10e.png