Your quality settings for Volumetrics and Volumetric Shadows are quite low. Especially Tile Size and Samples. Here is a comparison of your settings on the left and higher quality settings on the right:

This is only for a higher quality of the shadows. One of the reasons the cubes are not blocking enough light is not the fault of the cubes, but the brightness of the environment. Your world background is not very dark, and of course this lightens the volume and the shadowy parts of the scene as well. Here is a comparison between a brighter background and a darker background:

Of course the volume inside the spotlight cone will never look completely dark where it's shadowed by the cubes, because you are looking at the shadows through the surrounding volume that gets light. But one thing that can help increasing the contrast between light and shadow in the volume is the Anisotropy setting in the Volume Scatter which is kind of giving scattered light a preferred direction (simply spoken). Here a comparison between the default 0.0 value and 0.7 (but never use the maximum/minimum 1 and -1):

Last but not least - which doesn't matter in your scene since you have Soft Shadows disabled - what can make a difference is the Radius you set in the Light Properties. Your spotlight looks like it has a large radius (the circle around your light), maybe 1 m or 1.5 m, which can mean objects are not getting so much shadow because either the shadow is very soft or the object is even inside the light radius. So if you would have enabled Soft Shadows, here is a comparison of the shadows thrown by your light with e.g. 1.5 m radius on the left, and a smaller light with a radius of 0.01 m (like a small light bulb maybe) on the right:
