Long-subdivision extraction conjecture for digraphs

For a digraph $D$, write $\delta^+(D)$ for its minimum out-degree. A subdivision-path is a directed path replacing an arc in a subdivision.

Long-subdivision extraction conjecture. There is a function $f:\mathbb{N}\rightarrow\mathbb{N}$ such that for every $k\in\mathbb{N}$ and every digraph $D$ with

$$\delta^+(D)\geq f(k),$$

there exists a digraph $D'$ with $\delta^+(D')\geq k$ such that $D$ contains a subdivision of $D'$ in which every subdivision-path has length at least two.

The statement would imply the preceding subdivision-preservation conjecture, but is itself unresolved in the supplied text.