Long-time energy stabilization conjecture for hyperbolic obstacle problems

Let $s=1$ and consider an obstacle problem of the form of the wave equation with obstacle, with obstacle $g$, and let $u$ be the weak solution obtained through the convex minimization approach described in the paper. Write $E(u(t))$ for its energy. Long-time behavior conjecture. At least for sufficiently regular obstacles $g$, there exists $\bar{t}>0$ such that $E(u(t))$ is constant for every $t>\bar{t}$. The conjecture predicts that, after sufficiently long time, the solution no longer undergoes energy-dissipating impacts and its energy stabilizes; no proof or resolution is supplied in the paper.