Neither. Objectivism is perfectly compatible with SR and QM, and if an Objectivist claims otherwise, it proves only his own failure to adhere to the epistemology.
The universe is the way it is, and not how we might wish it to be.
I'd love to see any of the "Objectivist scientists" on this forum refute the claims made on the said website by either logical or scientific method.
He doesn't dispute the formalism, just the interpretation. I think his interpretation is lame, however. He simply postulates by fiat that rulers shrink and clocks slow down, and then insists that rulers and clocks don't "really" measure space and time, anyway. Nowhere does he say why, philosophically, I should expect the stub-your-toe physical reality of differently-aged twins. Nowhere does he describe a physical mechanism for atomic decays to occur at a slower rate. You can handwave away time dilation as a mere problem of perception all you want, but at the end of the day you are left with the reality these stone cold phenomena.