If you use different passwords, you must believe they are in different trust zones, and the usual time sharing methods won't work, in light of this and many other attacks. You will need separate CPUs as suggested by rayiner.
That’s the point: you can’t just pile every website into the same zone, and if you did pile them into the same zone, then using a different password for x.com and y.com is to some extent pointless because x can observe y’s password input and vice-versa.
(disregarding the fact that in current computing architectures “just put all the untrusted stuff on one core” is probably an insufficient solution due to shared caches anyways, no indeed nothing in this attack requires “sharing a core”)