Certainly, although you needn't have bothered with the terrible patronising greatly appreciated banoffee pudding and cream scones approach.
Because your system contains an AMD GPU.
I hope this helps.
If you have two 290s clearly PhysX isn't a big deal for you anyway, but I'll humour you.
http://www.nvidia.com/object/io_1202161567170.html
I don't see anything about AMD there. You have two AMD GPUs in your system. I'm going to use a software licence analogy here as it's probably the closest. If you have a licensed dongle for a software package. If you then go to use that dongle with another package, in this instance an AMD GPU (as you are using it as the main renderer in this instance which is without question of more importance). Then the licensed dongle will refuse to work. The fact you own another licensed dongle for the particular software you are trying to run, doesn't automatically mean that particular dongle should work for everything.
Given you're referring to the past 5 years, this should come as no surprise as PhysX is strictly an NV technology. You've paid for a theoretical dongle to render it on theoretically, a competitors package.
In any other situation, this would simply be tough ****. But because it's Nvidia, it's a terrible travesty. So you see it's quite difficult not to be slightly narky about it, when it's such a whine filled topic from the get go...
If some of you are truly hurt, I'd suggest maybe making a petition on Change.org and possibly get something done about it. Instead of moaning about Nvidia's ethics day in day out as if they owe you something for being such loyal customers