Not if they reduce it via the Misfortunes table! I always wondered how the Imperium could bleed you so thoroughly as to cause you to lose 1-5 PF in a single go through TAXES ALONE. I mean, what else could they be doing? Using your contacts? Looting your warehouses? Borrowing house guard and then getting them all killed? Fatally disrupt a trade route?
I Figured I'd start here and deal with all the comments below it: Misfortunes are only permanent if the RT ignores them. The examples presented would primarily (With the exception of the Administratum audit) only affect you if you were in fact the Imperial Governor of the world in question. Even so: An overzealous Logister chooses to inflict a Calamitous misfortune on the RT (1d5 PF he OF COURSE rolls a 5 ). In terms of my example, this is the equivalent of reducing your population from that of a Hive down to that of a City! That's pretty extreme I'd say!
The good news is it need not be permanent! By taking action, The RT can not only regain the lost influence/PF but can actually open up new opportunities for profit!
The Point behind all this is that when the Imperium takes over an area, It is unlikely they will displace the RT unless they've done something to Earn the Ire of the lords of Terra. Even if they do, The RT's business interests will remain and are fairly difficult to displace in any permanent way. Certainly Misfortunes can and do occur! But a Canny Lord Captain (Often with the aid of his Senechal!) can deal with this and even find opportunity in it!