Thomassen's connected partition conjecture with cross-degree condition

Let $k$ be a positive integer. A graph is $r$-connected if deleting fewer than $r$ vertices leaves it connected; for a vertex set $M\subseteq V(G)$, the condition that each vertex in $V_1$ has at least $k$ neighbours in $V_2$ is a cross-degree condition.

Thomassen's partition conjecture. For every $k\in\mathbb{N}$ there exists $f(k)\in\mathbb{N}$ such that if $G$ is an $f(k)$-connected graph and $M\subseteq V(G)$ consists of $k$ vertices, then there exists a partition $V_1,V_2$ of $V(G)$ such that $M\subseteq V_1$, both $G[V_1]$ and $G[V_2]$ are $k$-connected, and each vertex in $V_1$ has at least $k$ neighbours in $V_2$.

The source calls this a conjecture of Thomassen and explains that the graph version of the preceding tournament subdivision theorem would follow from it. Its resolution is not supplied in the paper.