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