The wall-dimension conjecture for TF equivalence classes

Let $A$ be a finite dimensional algebra over an algebraically closed field $k$. For each $\theta\in K_0(\operatorname{\mathsf{proj}} A)$, let $W_\theta$ be the associated subspace and let $|A|$ denote the number of isomorphism classes of simple $A$-modules. Let $\operatorname{{\rm ind}}\mathbb{N}\theta:=\bigcup_{\ell\ge1}\operatorname{{\rm ind}}\ell\theta$, where $\operatorname{{\rm ind}}\ell\theta$ is the set of canonical decomposition summands of $\ell\theta$.

Wall-dimension conjecture. For each $\theta\in K_0(\operatorname{\mathsf{proj}} A)$, we have

$$\dim_\mathbb{R} W_\theta=|A|-\dim\operatorname{cone}\nolimits(\operatorname{{\rm ind}}\mathbb{N}\theta).$$

The result preceding the conjecture gives equivalent criteria relating the dimension of $W_\theta$ to canonical decomposition data. The conjecture predicts the exact dimension for every class, extending the dimension bounds obtained from the linear independence of canonical summands.