Unavoidability of sufficiently deep outbranching balanced binary trees

An outbranching oriented tree with root $r$ is one in which, for every vertex $v$, the path from $r$ to $v$ is directed from $r$ to $v$. Let $B_d$ be the outbranching balanced binary tree on $2^{d+1}-1$ vertices in which every non-leaf vertex has two children as outneighbours and every leaf is at distance precisely $d$ from the root.

Balanced binary tree unavoidability conjecture. $B_d$ is unavoidable for sufficiently large $d$, possibly with $d>1$ sufficient.

This is proposed as a concrete case of the problem of finding sufficient conditions for an outbranching oriented tree to be unavoidable. The supplied text gives no resolution evidence beyond presenting it as a conjectural example.