Boston–Markin conjecture for totally real tamely ramified extensions

Let $G$ be a nontrivial finite group. Write $d_\lhd(G)$ for the least number of conjugacy classes in $G$ that generate $G$. A $G$-extension $K/\mathbb{Q}$ is a Galois extension with Galois group identified with $G$.

Boston–Markin conjecture. There exists a totally real tamely ramified $G$-extension $K/\mathbb{Q}$ such that the number of finite primes of $\mathbb{Q}$ ramified in $K$ is $d_\lhd(G)$.

The integer $d_\lhd(G)$ is a lower bound because the cyclic inertia subgroups generate $G$. The conjecture is known for abelian $G$ and remains open for every nonabelian $G$.