Suppose two $d$-dimensional r.v. $\xi,\eta$ have similar characteristic functions $\varphi_\xi(t)$ and $\varphi_\eta(t)$, like $$\sup_{t\in \mathbb R^d} |\varphi_\xi(t) - \varphi_\eta(t)| \leq \varepsilon$$ Can we bound the difference of cdf of the two r.v. when $x=0$ in terms of $\varepsilon$, i.e. $$ |F_\xi(0) - F_\eta(0)|\leq k\varepsilon $$ for some $k$ only depends on $d$?
If it helps, you can assume $\xi,\eta$ have compact support, and you can relax $k\varepsilon$ to any function of $\varepsilon$ that will tends to $0$ when $\varepsilon \to 0$, so long as it's independent to $\xi,\eta$.
Thanks!