Finite extremal conjecture for copies of rooted binary trees

Let $\operatorname{E}_n$ denote the rooted binary tree with $n$ leaves, and let $c(F,T)$ be the number of copies of a rooted binary tree $F$ in a rooted binary tree $T$. For integers $n\geq k$, consider all binary trees with $n$ leaves. Finite extremal conjecture. For every $n\geq k$, $\operatorname{E}_n$ has the largest number of copies of $\operatorname{E}_k$ among all binary trees with $n$ leaves. This is proposed as a finite analogue of the preceding asymptotic inducibility theorem; the source does not provide a resolution.