Freeness conjecture for the sharp transform of the planar binary tree algebra

Let ${\bf PBT}$ be the Loday–Ronco algebra of planar binary trees, and let ${\bf PBT}^\sharp$ denote its $\sharp$-transform. For each tree $T$ with at least two nodes and empty right subtree at the root, let ${\bf P}_T^\sharp$ be the corresponding sharp-transformed element.

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 the root is empty.

The claim describes a proposed free generating set for the sharp-transformed planar binary tree algebra. The supplied text gives no evidence that this assertion has been resolved.