Character classification for stable simple AF algebras

Let $A$ be a stable simple AF algebra not isomorphic to $\mathbb{K}$. Write $TW(A)$ for the tracial state space associated with $A$, $U_\to(A)$ for the relevant infinite-dimensional unitary group, and let $\operatorname{Char}(U_\to(A))$ denote its characters. For $\tau,\tau'\in TW(A)$ and $\varphi\in \operatorname{Hom}(K_0(A),\mathbb{Z})$, let $\mathrm{det}_\varphi\chi_{\tau,\tau'}$ be the corresponding determinant character. Character-classification conjecture.

$$\operatorname{ex}\operatorname{Char}(U_\to(A))=\{\mathrm{det}_\varphi\chi_{\tau,\tau'};\;\tau,\tau'\in TW(A),\;\varphi\in \operatorname{Hom}(K_0(A),\mathbb{Z})\}.$$

This asks whether the classification of indecomposable characters proved under the finite-dimensionality assumption on $TW(A)$ remains valid without that assumption; the source presents it as a natural question, and no resolution is supplied here.