They are explicitly not assuming anything about the content of the auxiliary space (full hard drive).
So the data might be incompressible and thus compressing it and restoring it afterwards would not work.
Edit:
From the paper:
> One natural approach is to compress the data on the hard
disk as much as possible, use the freed-up space for your computation and finally uncompress the data, restoring it to its original setting. But suppose that the data is not compressible. In other words, your scheme has to always work no matter the contents of the hard drive. Can you still
make good use of this additional space?
So the data might be incompressible and thus compressing it and restoring it afterwards would not work.
Edit: From the paper:
> One natural approach is to compress the data on the hard disk as much as possible, use the freed-up space for your computation and finally uncompress the data, restoring it to its original setting. But suppose that the data is not compressible. In other words, your scheme has to always work no matter the contents of the hard drive. Can you still make good use of this additional space?