Summary: Let \(F\) denote the class of all normalized analytic functions \(f\) that are locally univalent in the unit disk \(|z|<1\) satisfying the condition \[ \mathrm{Re}\left(1+\frac{zf''(z)}{f'(z)}\right)>-\frac{1}{2} \] for \(|z|<1\). Functions in \(F\) are known to be close-to-convex (univalent) in the unit disk. This class plays a crucial role in the discussion on certain extremal problems for the class of complex-valued and sense-preserving harmonic convex functions and some other related problems in determining univalence criteria for sense-preserving harmonic mappings. In this article, we show that every section of a function in the class \(F\) is convex in the disk \(|z|<1/6\). The radius \(1/6\) is best possible. We conjecture that every section of functions in the family \(F\) is univalent and close-to-convex in the disk \(|z|<1/3\).