  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 $\alpha_G(X)<\epsilon$.
