This is a follow-up to my previous question MO412306.
Let $X$ be a smooth complex projective surface of general type (this is the case I am mostly interested in, but one could ask the question in every dimension) and 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.
The nice answer to the aforementioned question shows that in this situation $K_X$ is ample, essentially because our assumption on $\Omega^1_X$ implies that $X$ does not contain any smooth rational curve. So, let me now ask the following
Question. Is $|K_X|$ base-point free? If not, what is a counterexample?
Remark. The answer is yes when $m=1$. In fact, if $\Omega_X$ is globally generated then, for every $x \in X$, we can find two global sections $a, \, b \in H^0(X, \, \Omega_X^1)$ such that $a(x), \, b(x)$ generate the fibre $\Omega^1_{X, \, x}$. Thus $a(x) \wedge b(x)$ generate $\wedge^2 \Omega^1_{X, \, x}=\omega_{X, \, x}$, namely, $a \wedge b$ is a global section of $K_X$ that does not vanish at $x$.
