McCuaig–Ota conjecture on contractible sets in 3-connected graphs

Let $G$ be a $3$-connected graph. A set $W\subset V(G)$ is contractible if $G(W)$ is connected and $G-W$ is $2$-connected; a contractible set with $k$ vertices is called a $k$-contractible set. McCuaig–Ota conjecture. For every $k\in\mathbb{N}$, there exists an integer $n$ such that every $3$-connected graph $G$ on at least $n$ vertices has a $k$-contractible set.

The conjecture asks whether sufficiently large $3$-connected graphs always contain contractible sets of every prescribed finite size. The paper proves the assertion for $k\geqslant 5$ under the minimum-degree condition $\delta(G)\geqslant \left[\frac{2k+1}{3}\right]+2$, while the unrestricted conjecture remains open in the supplied text.