The realization-map classification conjecture for Whitney tower filtrations

Let $\mathcal{T}_n$ be the tree group of order $n$, let $\text{\sf W}_n$ be the corresponding graded group of link concordance classes, and set $\widetilde{\mathcal{T}}_{2n}:=\mathcal{T}_{2n}$ while letting $\widetilde{\mathcal{T}}_{2n-1}$ denote the quotient by the framing relations. The realization maps are denoted by $R_n$. Realization-map classification conjecture. For every $n$, the realization maps induce isomorphisms

$$\text{\sf W}_n\cong\widetilde{\mathcal{T}}_n.$$

This would give a complete classification of the associated graded groups of the Whitney tower filtration, extending the established classifications in even orders and in orders congruent to $3\pmod 4$ after imposing the framing relations; the remaining cases are conjectural.