This question is on bounding the degree of the Todd class on a complex threefold.

Let $X$ be a smooth compact connected complex surface. Let $c_i=c_i(TX)$ be its $i$-th Chern class. Recall the following two facts. These allow one to bound the degree of the Todd class on a surface in terms of $c_2$.

**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, then the Bogomolov-Miyaoka-Yau inequality states that $$c_1^2 \leq 3c_2.$$

Now, I am interested in similar results for 3-dimensional smooth projective connected varieties over $\mathbb{C}$. In this case, the degree of the Todd class of $X$ is the degree of $$\frac{c_1 c_2}{24}.$$

Question. For 3-dimensional smooth projective connected varieties over $\mathbb{C}$, do there exist any absolute upper bounds on $c_1c_2$ (or any bounds for that matter) which are polynomial in $c_3$?