chern classes - Can one bound the todd class of a 3-dimensional variety polynomially in c_3

This question is on bounding the degree of the Todd class. Let me explain where this comes from.

Suppose that $X$ is a smooth compact connected complex surface. Let $c_i=c_i(TX)$ be its $i$-th Chern class.

1. If $X$ is not of general type, we have that $c_1^2$ is bounded absolutely from above by 9. See Table 10 of Chapter VI of Compact complex surfaces by Barth, Hulek, Peters and van de Ven.

2. If $X$ is of general type, the Bogomolov-Miyaoka-Yau inequality states that $$c_1^2 leq 3c_2.$$

Conclusion. Since the degree of the Todd class of $X$ is the degree of $$frac{c_1^2+c_2}{12},$$ we basically get an upper bound for this if we show that the degree of $c_2$ is bounded from above. Now, the degree of $c_2$ is the topological Euler characteristic. Thus, since my goal was to bound the Todd class from above (in the set-up of my problem), I reduce to bounding the topological Euler characteristic. (That's good!)

Now, I am really interested in a similar result for 3-dimensional smooth compact connected complex varieties.

In this case, the degree of the Todd class of $X$ is the degree of $$frac{c_1 c_2}{24}.$$ Since the topological Euler characteristic of $X$ is (the degree of) $c_3$, I would like to reduce the problem of bounding this quantity from above to bounding $c_3$. So now for my question.

Question. Do there exist any absolute upper bounds on $c_1c_2$ or any bounds for that matter which are polynomial in $c_3$?