Representative conjecture for generalized stationary space-times

Let $(\mathcal M,g)$ be a generalized stationary space-time, and let $(g_\varepsilon)_\varepsilon$ be a symmetric representative of its metric. The representative conjecture. On relatively compact open sets, one can choose $(g_\varepsilon)_\varepsilon$ so that $(\mathcal M,g_\varepsilon)$ is stationary, with Killing vector $\xi^a$, for every $\varepsilon>0$.

This would characterize generalized stationary space-times through stationary smooth representatives. The source gives no evidence that the assertion has been resolved.