The $4n^2$-inequality for complete intersection singularities

The famous $4n^2$-inequality is extended to generic complete intersection singularities: we show that the multiplicity of the self-intersection of a mobile linear system $\Sigma$ with a maximal singularity (i.e. the pair $(X,frac{1}{n}\Sigma)$ is not canonical, where $X$ is the ambient variety) is greater than $4n^2\mu$, where $\mu$ is the multiplicity of the singular point. This inequality essentially simplifies proving birational rigidity for many types of singular Fano varieties.