  In Part 1, we developed a new technique based on Lipschitz pushforwards for
proving the jump set containment property $\mathcal{H}^{m-1}(J_u \setminus
J_f)=0$ of solutions $u$ to total variation denoising. We demonstrated that the
technique also applies to Huber-regularised TV. Now, in this Part 2, we extend
the technique to higher-order regularisers. We are not quite able to prove the
property for total generalised variation (TGV) based on the symmetrised
gradient for the second-order term. We show that the property holds under three
conditions: First, the solution $u$ is locally bounded. Second, the
second-order variable is of locally bounded variation, $w \in
\mbox{BV}_\mbox{loc}(\Omega; \mathbb{R}^m)$, instead of just bounded
deformation, $w \in \mbox{BD}(\Omega)$. Third, $w$ does not jump on $J_u$
parallel to it. The second condition can be achieved for non-symmetric TGV.
Both the second and third condition can be achieved if we change the Radon (or
$L^1$) norm of the symmetrised gradient $Ew$ into an $L^p$ norm, $p>1$, in
which case Korn's inequality holds. We also consider the application of the
technique to infimal convolution TV, and study the limiting behaviour of the
singular part of $D u$, as the second parameter of $\mbox{TGV}^2$ goes to zero.
Unsurprisingly, it vanishes, but in numerical discretisations the situation
looks quite different. Finally, our work additionally includes a result on
TGV-strict approximation in $\mbox{BV}(\Omega)$.
