Non-rotationally invariant generalised geodesics in the disc

We study Brenier's relaxed least-action problem in the unit disc $D:=\{x\in\mathbb{R}^2:\ |x|<1\}$ at the critical time $T=π$, with the endpoint pair $i_D$ (the identity on $D$) and $-i_D$. We prove that the critical energy shell \[ S^3=\{(x,v)\in\mathbb{R}^2\times\mathbb{R}^2: |x|^2+|v|^2=1\} \] su