Restriction on minimum degree in the contractible sets problem

Let $G$ be a $3$-connected graph. A set $W\subset V(G)$ is called contractible if $G(W)$ is a connected graph and $G-W$ is a $2$-connected graph. In 1994, McCuaig and Ota conjectured that for any $k\in \mathbb{N}$ there exists $n\in \mathbb{N}$ such that any 3-connected graph $G$ with $v(G)⩾n$ has a $k$-vertex contractible set. It is proved that this holds if $k⩾5$ and $\delta (G)⩾\left[\frac{2k+1}{3}\right]+2$.