Demonet's lattice-point conjecture for brick-infinite algebras

Let $A$ be a finite-dimensional algebra over a field $K$. Write $K_0(\operatorname{\mathsf{proj}} A)$ for the Grothendieck group of finitely generated projective $A$-modules, and let $K_0(\operatorname{\mathsf{proj}} A)_\mathbb{R}$ denote its realification. A point in $K_0(\operatorname{\mathsf{proj}} A)$ is a lattice point, and a point $\theta$ is called rigid when it has the rigidity property used in the paper. The algebra $A$ is brick infinite if it has infinitely many isoclasses of bricks. Demonet's conjecture. If $A$ is brick infinite, then there exists a lattice point

$$\theta\in K_0(\operatorname{\mathsf{proj}} A)$$

which is not rigid. This strengthens the known characterization in which brick infiniteness is equivalent to the existence of a non-rigid point in $K_0(\operatorname{\mathsf{proj}} A)_\mathbb{R}$; the integral lattice-point assertion is stated as an open question.