The TF-class canonical decomposition conjecture

Let $A$) be a finite dimensional algebra over an algebraically closed field $k$. For $\theta\theta\in K_0(\operatorname{\mathsf{proj}} A)$, let $\theta=\theta_1\oplus\cdots\oplus\theta_\ell$ be a canonical decomposition, and define $\operatorname{{\rm ind}}\mathbb{N}\theta:=\bigcup_{\ell\ge1}\operatorname{{\rm ind}}\ell\theta$. Write $[\theta]_{\rm TF}$ for the TF equivalence class of $\theta$ and $\operatorname{cone}^\circ$ for the relative interior of a cone.

TF-class canonical decomposition conjecture. For each $\theta\in K_0(\operatorname{\mathsf{proj}} A)$, we have

$$[\theta]_{\rm TF}=\operatorname{cone}\nolimits^\circ(\operatorname{{\rm ind}}\mathbb{N}\theta).$$

The preceding theorem proves the corresponding inclusion with the right-hand side contained in the TF class. The conjecture asks for the reverse inclusion, and would describe every TF equivalence class through the canonical decomposition of the positive multiples of $\theta$.