Summary: We prove that the problem \[ \text{Minimize}\quad \int_ \Omega g(\Phi(\nabla T(x))) dx,\quad T\in T_ B+ W^{1, \infty}_ 0 (\Omega, \mathbb{R}^ n) \] admits at least one solution for any lower-semicontinuous extended-valued function \(g\), for any quasi-affine real-valued function \(\Phi\), and for any piecewise-affine boundary datum \(T_ B\) such that \(\Phi(\nabla T_ B)\) is constant.