In materials science and many other application domains, 3D information can often only be obtained by extrapolating from 2D slices. In topological data analysis, persistence vineyards have emerged as a powerful tool to take into account topological features stretching over several slices. It is illustrated how persistence vineyards can be used to design rigorous statistical hypothesis tests for 3D microstructure models based on data from 2D slices. More precisely, by establishing the asymptotic normality of suitable longitudinal and cross-sectional summary statistics, goodness-of-fit tests that become asymptotically exact in large sampling windows are devised. The testing methodology is illustrated through a detailed simulation study and a prototypical example from materials science is provided.