Could anesthesia cause progressive supranuclear palsy (psp)?

Not likely. No one knows for sure what causes it. It is a degenerative neurologic disease a little similar to Parkinson's disease in a comparable area of the brain. The cause may be genetic, viral, though no one really knows. It is very very rare.
PSP. There is zero evidence linking PSP to any form of anesthesia. Having said that it is also true that the cause(s) of PSP are unknown. Its virtually unheard of in people under 40. It is known that an abnormal protein (tau) builds up like some other neuro degenerative diseases. If you have PSP, you would be more at risk for other complications of anesthesia.