I am reading a paper$^\color{magenta}{\star}$ on $\ell_{1}$ trend filtering. In the conclusion, there is an optimization problem I am trying to solve. Assume we have data $y\in \mathbb{R}^{n}$ and we solve the following optimization problem
$$ (x,u) := \arg \min \frac12 \| y - x - u \|_{2}^{2} + \lambda \| D x \|_{1} + \rho \|u\|_{1},$$
where $D$ is trend filtering $\mathbb{R}^{(n-2)\times n}$ full row rank matrix. The problem is coercive, but strict convexity is not clear. This is called trend filtering with spikes.
I do not see why this problem is strictly convex, so that it always has a unique solution?
$\color{magenta}{\star}$ Seung-Jean Kim, Kwangmoo Koh, Stephen Boyd, Dimitry Gorinevsky, $\ell_1$ trend filtering, SIAM Review, 2009.