In-dominating-set conjecture for digraphs of large outdegree

For a digraph $D$, let $\delta^+(D)$ be its minimum out-degree. A set $X\subseteq V(D)$ is in-dominating if every vertex in $V(D)\setminus X$ has an out-neighbor in $X$.

In-dominating-set conjecture. There exists a function $f:\mathbb{N}\rightarrow\mathbb{N}$ such that, for every $k\geq1$, if

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

then there is an in-dominating set $X\subsetneq V(D)$ satisfying $\delta^+(D-X)\geq k$.

The source presents this as a natural statement that would imply the out-leaf extension conjecture; its resolution is not stated in the supplied text.