On estimates of the stability measure for decompositions of probability distributions into components

Let $\mathfrak G_m$ be the class of indecomposable probability laws with bounded spectrum $S(G)$ where
\begin{gather*} m=\min(u,v),\ u=G(\{\inf S(G)\}),\ v=G(\{\sup S(G)\}),\\ G(\{x\})=G(x+0)-G(x). \end{gather*}
If $G_1\ast G_2\in\mathfrak G_m$, $m>0$, $F_1$ has median 0 and if the uniform metric $\rho(F_1\ast F_2,G_1\ast G_2)\le\varepsilon$ then there exists a constant $\varepsilon_0=\varepsilon_0(G)>0$ such that
$$ \min\{\rho(F_1,G_1),\rho(F_1,G_2)\}\le(m-\sqrt{m^2-4\varepsilon})/2 $$
when $0\le\varepsilon\le\varepsilon_0$, and this estimate cannot be improved in the class $\mathfrak G_m$.