The decomposition conjecture for the twisted K-theory group of a regular automorphism

Let $A$ be a smooth $*$-algebra, let $\sigma$ be a regular automorphism of $A$, and let $F$ be its fixed point algebra. Let ${\mathcal S}$ be the semigroup of modular homotopy classes of unitaries over $A$ of the form $u_v$, where $v$ is a partial isometry over $A$ such that $vv^*$, $v^*v$, $v\sigma(v^*)$, and $v^*\sigma(v)$ all lie over $F$. Decomposition conjecture. There is an isomorphism

$$K_1(A,\sigma)\cong K_1(F)\oplus {\mathcal S}.$$

The conjecture is motivated by examples relating the twisted group $K_1(A,\sigma)$ to the $K$-theory of the associated fixed-point algebra and to modular homotopy classes. The source does not provide a resolution, so the conjecture remains open.