3 problems

Realization conjecture. There exists a tower of algebras $A_n$, with a basis $(e_\sigma)_{\sigma\in\mathfrak S_n}$, such that the restriction of $\rho_{m,n}$ to canonical words is…

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

Freeness conjecture. The algebra ${\bf PBT}^\sharp$ is free over the set ${\bf P}_T^\sharp$, where $T$ runs over trees with at least two nodes and such that the right subtree of th…