The extension conjecture for oriented trees

Let $D$ be a $k$-extension of an oriented tree of order $n$, meaning that there is a set of $k$ vertices whose deletion leaves the tree. Let $\operatorname{unvd}(D)$ denote unavoidability.

Extension conjecture. There is an absolute constant $C$ such that, for every integer $k$, every $k$-extension of an oriented tree of order $n$ is $C^k(2n-2)$-unavoidable; that is,

$$\operatorname{unvd}(D)\leqslant C^k(2n-2).$$

The source says this follows from the vertex-deletion conjecture together with Sumner's conjecture, so it is currently open with those conjectures. It concerns controlling the effect of adding finitely many vertices to an oriented tree.