  Let $\Gamma$ 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 $b_i\in \Gamma$, 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 $\Gamma$ 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 $E$.
