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It looks like this is a general problem after ICS upgrade for Sony Xperia phone.
I didn't have this problem before even in worse firmware 6.1.1.B.1.10, but after using 6.1.1.B.1.54 two weeks, I have the problem now...
I can connect to AP after boot.
After use for a little time, the process "wpa_supplicant" start running, and never stop, just like the screenshot below (right-top corner).
When the process "wpa_supplicant" is running, I can neither turn off the WiFi, nor make any setting change about WiFi.
What I can do is reboot.
It looks that the WiFi status was been locked by this process.
According the information on the web (reference), this issue can be fixed after delete the file /data/misc/wifi/wpa_supplicant_bagscan.db. (rooted required)
But this file will automatic generated by wpa_supplicant, so the same problem may happens again.
Is there any way to fix this issue without rooted?
This problem is solved in ICS c.1.10 and b.1.54 ... So just flash that firmwire, sony can't tell you to flash it manually ofcource... But, since b.1.54 isn't avaiable in all countries yet, you have to do it your self 
And NO, why the f. would you need a root for fixing wifi bug. That's nonsense. Rooting can't fix anything, just give you permisions to edit/move/delete system files.
So, get a bit of reason... If you have wifi bug, flash the latest firmwire for Xperia U (there are couple, depending on region as: Croatia,Nordic, etc...) And you're done 
I have the version 6.1.1.B.1.54 for a month. The phone works better, but this month I've had to do twice the repair because after a while it happens again. Do not know why, but I believe it has to do with the use of FON, or others that require authentication. Therefore, although better, the PROBLEM still exists.
I can confirm the problem does occur in 6.1.1.B.1.54.
My phone has gone back for repair now because of this problem but I've no reason to expect that it won't happen again sometime after it comes back.
Regarding the solution proposed by man1ac, I've rooted my Sony Xperia U (ICS), and have downloaded two different File managers, but when I go to the Data folder, there's no "wpa_supplicant_bgscan.db" file to be found... I've even tracked down the original post of HB78 to see if maybe I've missed something, but his directions are pretty much exaclty the same (as I can also speak/read Portuguese). I've also tried doing a manual search for this wpa_supplicant_bgscan.db file, but nothing can be found. Am I doing something wrong? I've been using the following path... (/My Files/mnt/sdcard/data)
Hello, Sony?
After 4 month, you still have any new information?
Do you already give up this phone?
Could please tell us the progress how you process this issue?
To Johan or any other Sony employee.
Can you confirm or deny in the change log for version B.89 firmware whether a fix for the wifi issue has been made and when will it be rolled out to PCC?
I've took my phone back due to the problem and have received a new Xperia U in return. If the problem has not been addressed then I will be selling the phone as new as I don't want the problem to come back as inevitably it will. I'm only prepared to keep it if it has been addressed/fixed.
To all those having problems I think the best way to prompt Sony about this is to return them to the point of purchase or manufacturer (depending on the sale of goods act in your country). If they start getting them returned then this may make them take their finger out.
Any update yet?
I have the same issue for months now (Xperia P) and am starting to get frustrated. Could Sony give more updates as to the current situation / status of the investigation? Also I don't think it is asked too much for a estimated date when this will be sorted by.
Currently it does give the impression that Sony focuses rather on new product development, than customer support!
For my part, this is putting me off future Sony devices (regardless of how nice the design is, etc), if this is the quality if their customer support.
Please sort this problem asap and give some updates, to show us that you are working on it!
I think Sony may NEVER fix this bug...
sony SOLVED this problem already.
I have just upgraded to 6.2.0.A.400 - and everything works !
I do not like sony, because they didnt answer and didnt tell me when they even PLAN to solve it - however this is solved now so i am happy.
Xperia U don't have this verson....
That's for Xperia P.