Realization conjecture for the planar binary tree Hopf algebra

Let $\mathfrak S_n$ be the symmetric group, let $(e_\sigma)_{\sigma\in\mathfrak S_n}$ be a basis of $A_n$, and let $\mathbf P_T$ denote the polynomial associated with a planar binary tree $T$ of size $n$. Let $\rho_{m,n}$ be the morphism associated with the tower, let $\mathcal K=\bigoplus_{n\geq 0}K_0(A_n)$, and define the induction product by

$$[M]\cdot[N]=[M\otimes_{\mathbb C}N\mathord\uparrow_{A_m\otimes A_n}^{A_{m+n}}].$$

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 given by the product of the corresponding $\mathbf P_T$ functions; the indecomposable projective modules of $A_n$ are the left ideals $P_T=A_ne_{\sigma_T}$ and are therefore in bijection with planar binary trees of size $n$; and the map

$$\mathcal K\longrightarrow\mathbf{PBT}$$

sending the class of $P_T$ to $\mathbf P_T$ is a ring isomorphism. This conjecture would provide an algebraic realization of the planar binary tree Hopf algebra, refining the existence conjecture and identifying its projective representation theory with the basis of planar binary trees. The source provides no evidence of resolution.