The τ-reduced second Brauer–Thrall conjecture

Let $A$ be a $\tau$-tilting infinite algebra. Write $\Irr^\tau(A,\bd)$ for the generically $\tau$-reduced irreducible components of the representation variety in dimension vector $\bd$, and let $c_A(\cZ)$ denote the codimension invariant associated with a component $\cZ$. A component is generically indecomposable when its general representation is indecomposable.

The $\tau$-reduced Brauer–Thrall II' conjecture. There exist $\bd\in\K_0(A)^+$ and a generically indecomposable $\cZ\in\Irr^\tau(A,\bd)$ such that

$$c_A(\cZ) \geq 1.$$

This is a geometric enlargement of the class of $\tau$-rigid modules to generically $\tau$-reduced components. The supplied source gives no resolution status.