  The notion of asymptotically log Fano varieties was given by Cheltsov and
Rubinstein. We show that, if an asymptotically log Fano variety $(X, D)$
satisfies that $D$ is irreducible and $-K_X-D$ is big, then $X$ does not admit
K\"ahler-Einstein edge metrics with angle $2\pi\beta$ along $D$ for any
sufficiently small positive rational number $\beta$. This gives an affirmative
answer to a conjecture of Cheltsov and Rubinstein.
