We present single-crystal high-pressure studies of the apohost of Dianin's compound (DC), as well as its clathrates with ethanol, acetic acid, and isopropanol, up to a pressure of 8.2(1) GPa. Despite having large cavities, these clathrates of DC are found to be very stable when exposed to high pressure, and even the apohost structure, which contains large void space regions, is stable up to pressures above 6.6(1) GPa. The similarity between the compression of the apohost and the DC clathrates is presumably due to the extraordinary strength of the host framework. Bulk modulus and linear compression values are derived for the apohost and DC/acetic acid clathrate. The observed compression is shown to be more isotropic than what is expected from experimental elastic tensor data in the literature, although the compression along the a-axis is the largest for all the studied clathrates. The pressure evolution of the refined atomic displacement parameters is shown to be a sensitive indicator of the accuracy of the crystal structure refinement.