Covering conjecture for generalized connected subsets over real closed fields

Let $R$ be a real closed field. For a smooth algebraic variety $X$ over $R$, write $X^{\operatorname{h}}$ for its associated space of real points. A generalized connected open subset is an open subset satisfying the three conditions in the definition above: it admits a smooth affine coordinate chart whose induced map is an open embedding on a neighborhood, its image is convex, and regular functions become convex on that image. Covering conjecture. The space $X^{\operatorname{h}}$ always admits a covering by generalized connected open subsets. This conjecture asserts a local convexity and connectedness property for smooth varieties over real closed fields, supporting generalized connectedness and Bertini-type results; the supplied text does not state whether it has been proved or disproved.