Image of the standardization of permutations of a finite multiset

Question

Let $M$ be a multiset $\{1^{m_1},2^{m_2},...\}$ whose cardinality $\#M:=m_1+m_2+...=:n$. Let $\Sigma:S_M\to S_n$ be the standardization map defined in Stanley combinatorics volume 1 ($S_M$ is the set of permutations of $M$). The author states in the solution of exercise 56 that $$\text{Im}(\Sigma)=\{\sigma\in S_n:D(\sigma^{-1})=\{m_1,m_1+m_2,...\}\cap [n-1]\}.$$ But this just seems wrong. Let $M=\{1^2,2^1,3^2\}$ so $\#M=5$. Then let $w\in S_M$ be $$w=12313$$ Applying standardization we get $$\Sigma(w)=13425$$ The inverse permutation of $\Sigma(w)$ is $14235$ and its only descent is $\{2\}$. I think that the condition should be relaxed to $D(\sigma^{-1})\subseteq\{m_1,m_1+m_2,...\}\cap [n-1]$