Let $X$ be an $n$-dimensional compact complex manifold with positive first Chern class. Are its Chern numbers $\prod_{i=1}^{n-1} c_i^{k_i}$, over $k _i \geq 0$ with $\sum ik_i = n$, bounded in terms of $c_n = (-1)^n e(X)$?

Remark. One may also include the convention $c_0 := 1$, for which the answer is positive since $c_n > 0$, in order to cover the prototypical observation that the Euler number of a hyperbolic surface is negative. In view of this example, I would also like to extend the original question to the realm of log manifolds and hyperbolicity.