The Gromov–Lawson–Rosenberg conjecture for totally non-spin spin^c manifolds

Let $(M,\sigma)$ be a closed connected totally non-spin spin$^c$ manifold with $\dim M=n$, fundamental group $\pi$, and classifying map

$$c\colon M\to B\pi.$$

Let $L$ be the associated spin$^c$ line bundle, let $A\colon K_*(B\pi)\to K_*(C^*_r(\pi))$ be the complex assembly map, and let $\operatorname{per}\colon ku_*(B\pi)\to K_*(B\pi)$ be the periodization map. Gromov–Lawson–Rosenberg conjecture, spin$^c$ version. For $n\geq 5$, $(M,\sigma,L)$ admits positive generalized scalar curvature if and only if

$$A\circ\operatorname{per}\circ c_*([M,\sigma])=0$$

in $K_*(C^*_r(\pi))$. This is the spin$^c$ analogue of the Gromov–Lawson–Rosenberg conjecture, with the higher index class of the Mishchenko–Fomenko spin$^c$ Dirac operator providing the obstruction. The supplied source does not state whether the conjecture is open or resolved.