Minimum degree conditions for containing an $r$-regular $r$-connected subgraph

We study optimal minimum degree conditions when an $n$-vertex graph $G$ contains an $r$-regular $r$-connected subgraph. We prove for $r$ fixed and $n$ large the condition to be $\delta(G) \ge \frac{n+r-2}{2}$ when $nr \equiv 0 \pmod 2$. This answers a question of M.~Kriesell.