Birational rigidity and alpha invariants of Fano varieties

We prove that for every $\epsilon >0$, there is a birationally super-rigid Fano variety $X$ such that $\frac{1}{2}⩽\alpha (X)⩽\frac{1}{2}+\epsilon$. Also we show that for every $\epsilon >0$, there is a Fano variety $X$ and a finite subgroup $G\subset \mathrm{A}\mathrm{u}\mathrm{t}(X)$ such that $X$ is $G$-birationally super-rigid, and ${\alpha }_G(X)<\epsilon$.