The rank-five K-theory conjecture for the algebraic noncommutative toroidal orbifold

Let $\mathcal A_\theta^{alg} \rtimes \mathbb Z_2$ be the algebraic noncommutative toroidal orbifold, and let $1,p^\theta,q_0^\theta,q_1^\theta,r^\theta$ be the five projections considered in the paper. Their classes are linearly independent in $K_0(\mathcal A_\theta^{alg} \rtimes \mathbb Z_2)$. The rank-five K-theory conjecture.

$$K_0(\mathcal A_\theta^{alg} \rtimes \mathbb Z_2) \cong \mathbb Z^5.$$

Equivalently, these five projection classes should span the $K_0$-group. Their linear independence follows from the corresponding independence in the smooth orbifold, while their spanning property is conjectural.