Hong, Kang and Yu's induced-cycle connectivity conjecture

Let $k$ and $l$ be positive integers. Define a graph to be $r$-connected if it remains connected after deletion of fewer than $r$ vertices. For a set $X$ of vertices, let $G-X$ be the graph obtained by deleting $X$, and let an induced cycle be a cycle with no edges between nonconsecutive vertices on the cycle. A set $X$ is incident with an edge $e$ if one of the endpoints of $e$ belongs to $X$.

Hong–Kang–Yu conjecture. There exists a smallest positive integer $g(k,l)$ such that, for every $g(k,l)$-connected graph $G$, every edge $e otin E(G[X])$ with $X e V(G)$ and $|X|=k$, there is an induced cycle $C$ in $G-X$ satisfying $e otin E(C)$ and $G-C$ is $l$-connected.

The conjecture asks for a uniform connectivity threshold guaranteeing a removable induced cycle while avoiding a prescribed set of vertices and preserving a prescribed edge condition. The surrounding text identifies it as a conjecture from Hong, Kang and Yu; no resolution is supplied here.