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The chromatic edge stability index $\mathrm{es}_{χ'}(G)$ of a graph $G$ is the minimum number of edges whose removal results in a graph with smaller chromatic index. We give best-possible upper bounds on $\mathrm{es}_{χ'}(G)$ in terms of the number of vertices of degree $Δ(G)$ (if $G$ is Class 2), and the numbers of vertices of degree $Δ(G)$ and ${Δ(G)-1}$ (if $G$ is Class 1). If $G$ is bipartite we give an exact expression for $\mathrm{es}_{χ'}(G)$ involving the maximum size of a matching in the subgraph induced by vertices of degree $Δ(G)$. Finally, we consider whether a minimum mitigating set, that is a set of size $\mathrm{es}_{χ'}(G)$ whose removal reduces the chromatic index, has the property that every edge meets a vertex of degree at least $Δ(G)-1$; we prove that this is true for some minimum mitigating set of $G$, but not necessarily for every minimum mitigating set of $G$.