On boundedness of divisors computing minimal log discrepancies for surfaces

Let $Γ$ be a finite set, and $X\ni x$ a fixed klt germ. For any lc germ $(X\ni x,B:=\sum_{i} b_iB_i)$ such that $b_i\in Γ$, Nakamura's conjecture, which is equivalent to the ACC conjecture for minimal log discrepancies for fixed germs, predicts that there always exists a prime divisor $E$ over $X\ni x$, such that $a(E,X,B)={\rm{mld}}(X\ni x,B)$, and $a(E,X,0)$ is bounded from above. We extend Nakamura's conjecture to the setting that $X\ni x$ is not necessarily fixed and $Γ$ satisfies the DCC, and show it holds for surfaces. We also find some sufficient conditions for the boundedness of $a(E,X,0)$ for any such $E$.

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