Freeness conjecture for the degree-one subalgebra of the noncommutative Connes–Kreimer algebra

Let $NCK$ be the noncommutative Connes–Kreimer Hopf algebra of Foissy, viewed as an algebra with two compatible associative products, and let $\fontfamily{U}\fontshape{n}\selectfont{As}^2$ denote the operad governing two compatible associative products. Consider the $\fontfamily{U}\fontshape{n}\selectfont{As}^2$-subalgebra of $NCK$ generated by elements of degree $1$, that is, by planar rooted trees with one leaf.

Freeness conjecture. The $\fontfamily{U}\fontshape{n}\selectfont{As}^2$-subalgebra of $NCK$ generated by elements of degree $1$ is a free algebra over the operad of two compatible associative products, one of which is in addition commutative.

The preceding discussion identifies commutativity of the product $cstar_2$ on this subalgebra as an obstruction to freeness as an algebra with two compatible products; the conjecture asserts that this is the only obstruction. The relevant operad is not Koszul, and little is known about the growth of the dimensions of its components.