Suppose that $K/\mathbb{Q}$ is a number field.
$(*)$ Let $L_1,\dots,L_n/K$ be a finite family of finite Galois extensions such that $L_i\cap L_j = K$ for all $i,j$.
Is this enough to conclude that $L_1,\dots,L_n$ are linearly disjoint over $K$, i.e. $$ L_j\cap L_1\dots L_{j-1}=K \quad\text{for all }j=1,\dots,n? \tag{**} $$ Clearly, the $(**)$ implies $(*)$. But I believe $(*)$ implies $(**)$ is false in general. Does anyone have a counterexample that they can share with me?