The cherry-count conjecture for random tanglegrams

A cherry is a pair of leaves joined to a common parent in a binary tree. Let a tanglegram of size $n$ be chosen uniformly at random, and consider its left binary tree. Cherry-count conjecture. The expected number of cherries in the left tree converges to $n/4$ as $n$ tends to infinity. This predicts linear growth for the expected number of cherries in a random tanglegram; the source presents it as suggested by computer experiments and does not state a proof.