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The notion of asymptotically log Fano varieties was given by Cheltsov andRubinstein. 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 admitK\"ahler-Einstein edge metrics with angle $2\pi\beta$ along $D$ for anysufficiently small positive rational number $\beta$. This gives an affirmativeanswer to a conjecture of Cheltsov and Rubinstein.

On log K-stability for asymptotically log Fano varieties