Constant concentration for chi square variables

Question

I would like to have a certain concentration inequality for chi square variables with $k$ degrees of freedom. As referenced in https://stats.stackexchange.com/questions/4816/what-are-the-sharpest-known-tail-bounds-for-chi-k2-distributed-variables, Laurant and Massart give as a corollary that $$ \Pr\left(\left\lvert X-\mathbb E(X)\right\rvert\geq 2\sqrt{kx}+2x\right)\leq 2\exp(-x). $$ However, I only care about the case where the right hand side is constant, say $1/10$ (so $x\approx 3$), and I would like a weaker dependence on $k$, say $k^{1/4}$ instead of $k^{1/2}$. I only care about the asymptotic order of $k$, so anything $O(k^{1/2-\varepsilon})$ would be great. Could anyone suggest any results/approaches, or could it be impossible? Thank you!