Weak lower semicontinuity conjecture for homogenized elliptic energies

Let $\Omega$ be the domain, let $A_{\varepsilon}$ and $B_{\varepsilon}$ be the coefficient fields from the preceding boundary-value problem, and let $B^\sharp$ be the homogenized coefficient associated with $B_{\varepsilon}$. For weakly convergent data $g_{\varepsilon}\rightharpoonup g$ in $H^{-1}(\Omega)$, let $v_{\varepsilon}\in H^1_0(\Omega)$ be the weak solution of

$$-\operatorname{div}(A_{\varepsilon}\nabla v_{\varepsilon})=g_{\varepsilon}\quad\text{in }\Omega,\qquad v_{\varepsilon}=0\quad\text{on }\partial\Omega,$$

and suppose that $v_{\varepsilon}\rightharpoonup v_0$ weakly in $H^1_0(\Omega)$, where $v_0\in H^1_0(\Omega)$ is the unique solution of the homogenized limit problem. Weak lower semicontinuity conjecture. One has

$$\liminf_{\varepsilon\to 0}\int_\Omega B_{\varepsilon}\nabla v_{\varepsilon}\cdot\nabla v_{\varepsilon}\,dx\geq\int_\Omega B^\sharp\nabla v_0\cdot\nabla v_0\,dx.$$

The conjecture asks whether the homogenized energy gives a lower bound for sequences of solutions with weakly convergent right-hand sides; the source states that it is open in general.