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Take a unit sphere $S^2$ and a hyperbolic surface $X$. Then the product $S^ 2 \times X$ is not flat and has zero scalar curvature. Also it is conformally flat by a paper

Simon Salamon: Complex structures and conformal geometry. In: BOLLETTINO DELLA UNIONE MATEMATICA ITALIANA, vol. 9, pp. 199-224

as per editor Holonomia.

Take a unit sphere $S^2$ and a hyperbolic surface $X$. Then the product $S^ 2 \times X$ is not flat and has zero scalar curvature. Also it is conformally flat by a paper

Simon Salamon (2009) Complex structures and conformal geometry. In: BOLLETTINO DELLA UNIONE MATEMATICA ITALIANA, vol. 9, pp. 199-224

as per editor Holonomia.

Don't have a solution at this point.

Take a unit sphere $S^2$ and a hyperbolic surface $X$. Then the product $S^ 2 \times X$ is not flat and has zero scalar curvature. Also it is conformally flat by a paper

Simon Salamon: Complex structures and conformal geometry. In: BOLLETTINO DELLA UNIONE MATEMATICA ITALIANA, vol. 9, pp. 199-224

as per editor Holonomia.

The previous answer was not correct. For a conformally flat metric the sectional curvature in every two-dimensional direction is expressed in terms of the laplace-beltrami operator of the log of the conformal factor. Therefore the scalar curvature is proportional to the sectional curvature and so the manifold has to be flat. This is undergraduate differential geometry that I have taught many times, sorry.

Don't have a solution at this point.