On the existence of minimizer on a log Fano cone singularity

We prove that if $x\in (X,\mathrm{\Delta },\mathbb{T})$ is a log Fano cone singularity over an uncountable algebraically closed field, and ${\nu }_0$ is a $\mathbb{T}$-invariant valuation with center $x$ and $A_{X,\mathrm{\Delta }}({\nu }_0)<\mathrm{\infty }$, then the value $$\delta (X,\mathrm{\Delta };{\nu }_0):=\underset{\nu \in {\mathrm{V}\mathrm{a}\mathrm{l}}_{X,\ni x}^{\mathbb{T},\ast }}{\inf }\frac{A_{X,\mathrm{\Delta }}(\nu )}{S({\nu }_0;\nu )}$$ admits a minimum. The proof uses the generic limit argument. Note that there is a counterexample if we discard the log Fano cone structure.