Let $M$ be a complex manifold with real tangent bundle $TM$. Let $J$ be its associated almost complex structure ($J\circ J=-\operatorname{id}$) and $\nabla$ a torsion free, flat connection in $TM$ compatible with $J$, that is $$\nabla J=0.$$

Is it true that in this case the manifold is geodesically complete?

If one replaces the complex structure by a symplectic structure, the question becomes a simpler version of the Markus conjecture.