A local estimate for vectorial total variation minimization in one dimension

Let $\boldsymbol u$ be the minimizer of vectorial total variation ($VTV$) with $L^2$ data-fidelity term on an interval $I$. We show that the total variation of $\boldsymbol u$ over any subinterval of $I$ is bounded by that of the datum over the same subinterval. We deduce analogous statement for the vectorial total variation flow on $I$.