Let $K=\mathbb{Q}(\sqrt{-q})$ where $q\equiv 7\bmod 16$ is a prime. Take $F=K(\sqrt[4]{-q},\sqrt{-1})$. Then we proved that $X_{F\tilde{K}}=0$; see the [preprint on arXiv](https://arxiv.org/abs/2008.10310). In particular, $X_{T\tilde{K}}=0$ for every intermediate field $T$ in the extension $F/K$.