The Enhanced X-ray Timing and Polarimetry (eXTP) mission is a space mission to be launched in the late 2020s that is currently in development led by China in international collaboration with European partners. Here we provide a progress report on the Czech contribution to the eXTP science. We report on our simulation results performed in Opava (Institute of Physics of the Silesian University in Opava) and Prague (Astronomical Institute of the Czech Academy of Sciences), where the advanced timing capabilities of the satellite have been assessed for bright X-ray binaries that contain an accreting neutron star (NS) and exhibit the quasi-periodic oscillations. Both stellar-mass, supermassive, and tentative intermediate-mass black holes can be included in the program portfolio. Measurements of X-ray variability originating in oscillations of fluid in the innermost parts of the accretion region determined by general relativity, such as the radial or Lense-Thirring precession, can serve for sensitive tests enabling us to distinguish between the signatures of different viable dense matter equations of state. We have developed formulae describing non-geodesic oscillations of accreted fluid and their simplified practical forms that allow for an expeditious application of the universal relations determining the NS properties. These relations, along with our software tools for studying the propagation of light in strong gravity and neutron star models, can be used for precise modeling of the X-ray variability while focusing on properties of the intended Large Area Detector (LAD). We update the status of our program and set up an electronic repository that will provide simulation results and gradual updates as the mission specifications progress toward their final formulation.