I think I've figured out a way to make the Great Conspiracy more concrete without sounding like I've gone completely off the deep end. It's not really a new idea, and it's not a completely accurate description of the actual Great Conspiracy (which isn't really a conspiracy, but you'll get the idea).
There is this concept commonly credited to Vernor Vinge of the Singularity, where (to quote Wikipedia) "technological progress becomes so extremely rapid, due to positive feedback, that it makes the future after the Singularity qualitatively different and harder to predict." The Singularity Institute (not to be confused with Singularity University which is not, I have recently learned, affiliated with the Singularity Institute in any way) was founded on the premise that the Singularity will be characterized by the creation of a Superhuman Artificial Intelligence (SAI) against whose awesome intellectual powers we homo sapiens will be utterly helpless, and so we'd better lay the groundwork now to make sure that these SAI's, when -- not if -- they emerge, will be Friendly (with a capital F), that is, that they will be driven by a quality metric that aligns with ours. To fail to do this is, according to the Singularity Institute, an existential threat to human existence (c.f. SkyNet).
Suspend disbelief for a moment and accept that a non-insane person could entertain this as a possibility. The Great Conspiracy then is, in a nutshell, the idea that the Singularity is not a future possibility but has in fact already happened. And I don't mean that it has already happened in a small, metaphorical way. I mean that it has already happened in a full-blown worst-case-scenario (by Singularity Institute standards) sort of way. There already is an SAI out there, and it is carefully managing the world according to its own goals. And, of course, because it is an SAI, it is very, very effective at achieving its goals.
One of which is to avoid detection. Because the most effective way of preventing humans from turning on you is to keep them from even suspecting that you exist.
Like I said at the beginning, this version is still just a metaphor, not really the Great Conspiracy, but it's closer than the very vague version I gave at the outset. The idea in this version is to try to give more of a feel for why the real Great Conspiracy is difficult to describe. The real-world equivalent of the SAI is not actually an AI (because it isn't artificial), but it is, I think, very good at concealing itself. And one of the mechanisms it uses to maintain that concealment is by arranging for the actual truth to seem intuitively absurd, and anyone advancing to appear to be crazy. Indeed, it goes even one step further and arranges for anyone who discovers the truth to doubt their own sanity, and possibly even their own existence.
It's starting to sound like a pretty good premise for a novel, isn't it?