KAMAK tree conjecture for grounded forests

Let a digraph $D$ have a height function be a mapping $h:V(D)\rightarrow\mathbb{Z}$ such that $h(v)=h(u)+1$ for every arc $(u,v)\in A(D)$. An oriented forest is grounded if it admits a height function constant on the set of vertices of in-degree at least $2$. A digraph $F$ is $\delta^+$-enforcible if there exists $d(F)\in\mathbb{N}$ such that every digraph with minimum out-degree at least $d(F)$ contains a subdigraph isomorphic to $F$.

KAMAK tree conjecture. Every grounded forest is $\delta^+$-enforcible.

Hons et al. proved the necessary condition that every $\delta^+$-enforcible digraph is a grounded forest. The conjecture asserts sufficiency and is the main result proved in the paper, although the supplied status metadata marks it as open.