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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 paper (https://link.springer.com/article/10.1007/s00029-021-00644-3). In particular, $X_{T\tilde{K}}=0$ for every intermediate field $T$ in the extension $F/K$.