Let $f : \mathbb{R}^n \rightarrow \mathbb{R}_+$ be a convex, increasing, and continuously differentiable function, define an implicit function $g : \mathbb{R}_+ \rightarrow \mathbb{R}_+$ in the following manner: $$ g(l) = \| \nabla f(x_l)\|_2,$$ where \begin{align} x_l = \arg&\min_x \frac{1}{2} \| x - y\|_2^2 \\ \text{s.t.} \quad & f(x) \le l. \end{align} Intuitively, $x_l$ is the projection of $y$ onto the sublevel set $ K_l = \{ x \mid f(x) \le l\}$. I would like to show that the function $g(l)$ is non-decreasing, or come up with counter example that this is not true. This is trivially true when $n = 1$, but I am not sure how to show it in higher dimensions.
I have tried using the first order condition: $$ y - x_l = \lambda_l \nabla f(x_l), $$ where $\lambda_l$ is the optimal dual variable, hence $\|\nabla f (x_l)\| = \frac{1}{\lambda_l}\| y - x_l \|$, but it doesn't seem to help.