Local induced-cycle conjecture for vertex-prescribed deletion

A graph is $r$-connected if deleting fewer than $r$ vertices leaves it connected. An induced cycle is a cycle with no edges between nonconsecutive cycle vertices.

Vertex-prescribed induced-cycle conjecture. For every $k$, there exists an $h(k)$ such that for every vertex $v$ of every $h(k)$-connected graph $G$, there exists an induced cycle $C$ containing $v$ such that $G-V(C)$ is $k$-connected.

This is a weaker local form of the high-connectivity-keeping cycle problem, while the corresponding edge-prescribed form is equivalent to the high-connectivity-keeping path conjecture. The source states that this problem is open.