Cha's primary decomposition conjecture for the topologically slice concordance group

Let $\mathcal{Q} \subset {\mathbb Z}[t]$ be the set of irreducible polynomials $q(t)$ satisfying $q(1)=1$, and let

$$\mathcal{P}=\{q(t)q(t^{-1})\in {\mathbb Z}[t,t^{-1}]\mid q(t)\in\mathcal{Q}\}.$$

Let $\mathcal{T}$ be the subgroup of the smooth knot concordance group consisting of topologically slice knots. For $p\in\mathcal{P}$, let $\mathcal{T}^p$ be the subgroup generated by topologically slice knots whose Alexander polynomial is a product of copies of $p$, and let $\mathcal{T}^1$ be the subgroup generated by topologically slice knots with Alexander polynomial one. Define

$$\mathcal{T}^p_\Delta=\mathcal{T}^p/\mathcal{T}^1,\qquad \mathcal{T}_\Delta=\mathcal{T}/\mathcal{T}^1.$$

Cha's primary decomposition conjecture. The canonical homomorphisms $\mathcal{T}^p_\Delta\to\mathcal{T}_\Delta$ induce an isomorphism

$$\Phi:\bigoplus_{p\in\mathcal{P}}\mathcal{T}^p_\Delta\to\mathcal{T}_\Delta.$$

This conjecture asserts that, after quotienting by the subgroup generated by Alexander-polynomial-one knots, the topologically slice concordance group decomposes into independent primary summands indexed by the symmetric Alexander polynomials in $\mathcal{P}$. The statement is presented here without evidence of a resolution.