On the cohomological dimension of some pro-$p$-extensions above the cyclotomic $\mathbb Z_p$-extension of a number field

Let $\widetilde K_S^T$ be the maximal pro-$p$-extension of the cyclotomic $\mathbb Z_p$-extension $K^\mathrm{cyc}$ of a number field $K$, unramified outside the places above $S$ and totally split at the places above $T$. Let $\widetilde G_S^T=\mathrm{Gal}(\widetilde K_S^T/K)$.
In this work we adapt the methods developed by Schmidt in order to show that the group $\widetilde G_S^T=\mathrm{Gal}(\widetilde K_S^T/K)$ is of cohomological dimension 2 provided the finite set $S$ is well chosen. This group $\widetilde G_S^T$ is in fact mild in the sense of Labute. We compute its Euler characteristic, by studying the Galois cohomology groups $H^i(\widetilde G_S^T,\mathbb F_p)$, $i=1,2$. Finally, we provide new situations where the group $\widetilde G_S^T$ is a free pro-$p$-group.