Let $X$ be a smooth complex projective variety *of general type*; in my applications, I work with a surface, but let me ask this question in full generality.

Assume that for some $m \geq 1$ the vector bundle $S^m \Omega_X^1$ is generated by global sections, namely, the evaluation map $$H^0(X, \, S^m \Omega_X^1) \otimes \mathcal{O}_X \to S^m \Omega^1_X$$ is surjective.

Question.Is it true that $K_X$ is ample? Otherwise, what is a counterexample?

I started working on these topics rather recently, so I apologize if this question turns out to be trivial for the experts. Any answer and/or reference to the relevant literature will be highly appreciated.

**Edit** (12/26/2021). Follow-up question about the base-point freeness of $|K_X|$ asked as MO412382.