The equivalence of avoiding and enumerating generating functions for binary tree patterns

Let $s$ and $t$ be binary tree patterns. They are avoiding-equivalent if their avoiding generating functions satisfy

$$\operatorname{Av}_s(x)=\operatorname{Av}_t(x),$$

and they are enumerating-equivalent if

$$\operatorname{En}_{L,s}(x,y)=\operatorname{En}_{L,t}(x,y).$$

Avoiding-enumerating equivalence conjecture. If $s$ and $t$ are avoiding-equivalent, then they are also enumerating-equivalent.

This conjecture asserts that equality of the numbers of binary trees avoiding two patterns forces equality of the refined generating functions recording the number of pattern copies. The paper reports computational evidence for binary tree patterns with at most $7$ leaves, but does not establish the assertion in general.