It should--I don't actually remember testing it on 4.0.1, but I'm pretty certain that it uses the same run-as file as 4.0.2 in which the method exploits.
If you've still got 4.0.1, this should work for you.
~ ~ ~
By the way, there's a
new root without unlocking the bootloader method:
http://forum.xda-developers.com/showthread.php?t=1894717.
I'm not 100% sure which versions of JB and ICS this applies to--you'll have to dig in to the thread for details (I haven't finished reading it yet).
Cheers and best of luck!