The Abyssals aren't dead, but they do have a thing for serving ghosts, creating zombies and trying to spread the taint of the Underworld throughout Creation. It's kind of their raison d'être. No, that doesn't mean that every Liminal will automatically try to kill every Abyssal on sight, but it means that Abyssals, being the only Exalt with a specific schtick of raising the dead and leading armies of ghosts to conquer the lands of the living, are the only Exalt that Liminals have a specific reason to oppose.
I'm not talking about power tiers, I said numerous before powerful for a reason. There just aren't enough Liminals for the Abyssals as a whole to care. But yes, power does matter in this case. The Liminals are individually weak compared to Abyssals, even though being weaker doesn't mean they autolose to all Abyssals forever, which means that, combined with their very low numbers, no matter how much the Liminals may or may not hate the Abyssals, it is reasonable for a given Abyssal to simply not care about Liminals or even know what they are. They can ignore the existence of Liminals in a way that a Solar could never ignore the existence of the Dragon-Blooded (because they're everywhere and they collectively are the most powerful Exalt) or a Dragon-Blooded can't ignore the existence of Solars or Lunars (because their dark powers and the sheer level of threat they pose on an individual level makes killing them one of the main purposes, societally speaking, of having the Dragon-Blooded, and on the occasions that you do encounter them it's a Huge Deal rather than an unusually strong enemy). If you can casually ignore the other splat's very existence and nothing will change as a result of that, they can hardly be considered your One True Enemy.