Let $X$ be a projective normal variety, $D$ be a Cartier divisor on $X$ and $A$ be an ample divisor on $X$. Let $x \in X$ be a (not necessarily closed) point. If the asymptotic vanishing order of $D$ (which is defined to be the infimum of the vanishing order of all Q-divisors which Q-linearly equivalent to $D$) at every point in a neighborhood $U$ of $x$ is zero, then does there exist a general Q-divisor $D'$ which is Q-linearly equivalent to $D+A$ and the multiplicity of $D'$ at $x$ is zero?