Let $X_1,\dots,X_n$ denote i.i.d. standard Gaussian random variables. My question is: Do there exist absolute constants $C,c>0$ such that for every $\epsilon>0$ and positive real numbers $a_1,\dots,a_n>0$, we have the following bound

$\quad \Pr(\sum_{i=1}^n a_i X_i^2 \le \epsilon \sum_{i=1}^n a_i)\le C\epsilon^c\quad ?$

Background: Results of Carbery and Wright (2001) give powerful anti-concentration inequalities for polynomials of Gaussian variables. If we replace the term $\epsilon \sum_{i=1}^n a_i$ by $\epsilon\sqrt{\sum_{i=1}^n a_i^2}$ the bound follows directly from Carbery-Wright with $c=1/2$. Here, however, we need a stronger bound in terms of the $\ell_1$-norm rather than $\ell_2$-norm of the polynomial.

The reason why I believe the claim is still true is that all the variables $X_i$ appear squared so that there are no cancellations. Also, for $n=1$ the claim follows from simple Gaussian anti-concentration bounds with $c=1/2.$

Note: A simple application of Paley-Zygmund gives the bound $\Pr(\sum a_i X_i^2 \ge \epsilon\sum a_i )\ge (1-\epsilon)^2/3.$ This, unfortunately does not imply the statement above.