Periodic-cyclic-homology formula for the stringy Euler characteristic

Let $X$ be a normal Gorenstein variety with a noncommutative crepant resolution (NCCR). For an algebra or sheaf of algebras $\Lambda$, define its Euler characteristic by

$$e(\Lambda)=\dim\operatorname{HP}^{\mathrm{even}}(\Lambda)-\dim\operatorname{HP}^{\mathrm{odd}}(\Lambda),$$

where periodic cyclic homology is computed for a dg enhancement when necessary. Let $e_{st}(X)$ denote the stringy Euler characteristic.

Stringy Euler-characteristic conjecture. The stringy Euler characteristic of $X$ can be computed as the Euler characteristic of an NCCR, computed via periodic cyclic homology.

For a commutative crepant resolution, the analogous equality follows from Hochschild–Kostant–Rosenberg and the equality of stringy and ordinary Euler characteristics. The source explicitly warns that the assertion does not extend to twisted NCCRs.