Écalle–Chapoton conjecture on L-classes in planar binary trees

Let ${\bf PBT}$ be the space of planar binary trees, let ${\rm Lie}$ be the Lie subspace, and let the L-classes be the classes of binary trees described in the source. A Lie element of ${\bf PBT}$ is an element of ${\bf PBT}\cap{\rm Lie}$.

Écalle–Chapoton conjecture. Any Lie element of ${\bf PBT}$ is a linear combination of L-classes.

The L-classes arise by grouping binary trees that have the same coefficient in a generic linear combination constrained to belong to ${\bf PBT}$. The source notes that there are more L-classes in size $n$ than the dimension of ${\bf PBT}_n\cap{\rm Lie}(n)$, so the conjecture gives only a necessary condition.