The generator-count asymptotic conjecture for random tanglegrams

Let $B_n$ denote the set of binary trees with $n$ leaves. For a binary tree $T$, let $l(T)$ be its number of leaves and $k(T)$ its number of symmetries, meaning vertices with identical subtrees. Let $T'\in B_n$ be the left tree of a uniformly random tanglegram, and let $A(T')$ be the associated object whose generators are being counted. Generator-count conjecture. The expected number of generators of $A(T')$ is asymptotically equal to

$$\left(\sum_{T\in B_n}\frac{1}{4^{l(T)+k(T)}}\right)n.$$

The claim is presented as a consequence that would follow if the preceding subtree-count conjectures hold; the source does not establish it.