Jacobian of Proximal Operator is Positive Semidefinite

Question

I have been reading a paper involving the following proximal operator $\hat{y}:R^{p} \to R^p$: $$\hat{y}(v) := \text{argmin}_{\beta \in {R}^{p}} \left\{\frac{1}{2} \|v - \beta\|_2^2 + \theta \|\Sigma^{-1/2} \beta \|_1 \right\}$$ where $\theta > 0$ is a constant and $\Sigma$ is positive definite matrix. According to the paper, "by general properties of proximal operators, the Jacobian matrix is positive semi-definite." The paper refers to the book "Convex Analysis and Monotone Operator Theory in Hilbert Spaces" by Heinz H. Bauschke and Patrick L. Combettes. However, I was unable to find such results in the book. I can prove this result when $\Sigma = I_p$ because the proximal operator has close form. It is not clear to me whether it is correct for general $\Sigma$. Could anyone please help me with this? Thanks a lot!