The non-decreasing condition conjecture for generically tau-reduced components

Let $\Lambda$ be the finite-dimensional algebra under consideration, and let

$$\mathcal{Z}=\overline{\mathcal{Z}_1\oplus\cdots\oplus\mathcal{Z}_s}$$

be a generically $\tau$-reduced component, with $|\Lambda|$ denoting the number of isomorphism classes of simple $\Lambda$-modules and $|\mathcal{Z}|$ the number of indecomposable components occurring in $\mathcal{Z}$. The non-decreasing condition conjecture. The following statements hold:

1. $|\mathcal{Z}|\le |\Lambda|$.
2. $|\mathcal{Z}|=|\Lambda|$ if and only if, for every $1\le i\le s$,

$$\min\{\operatorname{hom}_{\Lambda}(X,\tau X)\mid X\in\mathcal{Z}_i\}=0,$$

and $\mathcal{Z}$ is maximal: whenever $\mathcal{Z}'$ and $\overline{\mathcal{Z}\oplus\mathcal{Z}'}$ are generically $\tau$-reduced components, one has

$$|\mathcal{Z}|=|\overline{\mathcal{Z}\oplus\mathcal{Z}'}|.$$

The result is presented as the main conjectural statement studied under the non-decreasing condition on a g-vector; the surrounding discussion indicates that the paper proves the asserted bounds and characterization under that condition. Its broader status without the non-decreasing hypothesis should be checked against the paper's later results.