 Solution

Suppose ${\hat{\beta }}_1$ and ${\hat{\beta }}_2$ minimize $Q$. Their midpoint is also a minimizer because $Q$ is a convex function. The group penalty is convex, while the squared Euclidean norm is strictly convex in the fitted value. If $X{\hat{\beta }}_1\ne X{\hat{\beta }}_2$, strict convexity would make the midpoint objective strictly smaller than the minimum. Hence $X{\hat{\beta }}_1=X{\hat{\beta }}_2$, so the fitted values are unique.