E.G.

A standard (I think) story: Before the chemical composition of water was known, people didn't know, and couldn't know, that water=H2O. But they still had the (or a) concept of water. It's just that their concept of water didn't include the fact that water=H2O. So they could deny that water=H2O without thereby manifesting conceptual incompetence. After the science of chemistry got going full steam, evidence that water=H2O accumulated, and then became decisive, and water=H2O became settled science, and eventually, water=H2O found its way into our shared understanding of what water is. So now it is the case that if you deny that water=H2O, your grasp of the concept of water is thereby revealed to be partial at best.

If we're telling this story, then I suppose we should allow that false propositions can be conceptually necessary. Here are two stories to think about:

Story #1: As above, except that the evidence for water=H2O was misleading all along. The truth is that water=XYZ. There's actually no such thing as hydrogen or oxygen, so no such thing as H2O. But there is such a thing as water, and water=XYZ. 

Story #2: The history of chemistry goes differently. People come to believe, falsely, that gold=lead, and this idea becomes so widely accepted that it finds its way into our shared understanding of what gold is.

If we think that, in general, either

(a) P is conceptually necessary if denial of P manifests conceptual incompetence,

or

(b) P is conceptually necessary if there is some concept that guarantees that P,

then it seems to me that we should say that in Story #1, water=H2O is false but conceptually necessary, and that in Story #2, gold=lead is false but conceptually necessary.

***

A moral fixed points-related idea is that some moral propositions are both substantive and conceptually necessary. Let's say that such propositions are moral-conceptual freebies. Here's a claim to consider:

Freebie Factivity: Moral-conceptual freebies are always true.

If Freebie Factivity were true, it would mean either that conceptually necessary falsehoods don't exist anywhere (which I doubt, because of Story #1 and Story #2 above), or that morality is special, such that none of the conceptually necessary falsehoods are moral propositions.

It would be surprising if morality were special in that way. Indeed, if there are any conceptually necessary falsehoods at all, then moral thought might be particularly rife with them. Societies, institutions, and individuals might have both the ability and the incentive to distort moral concepts in ways that serve their interests or reflect their biases.

Here's a claim that I think probably is a moral-conceptual freebie (if there are any moral-conceptual freebies):

(1) It is morally wrong to torture an innocent child just for fun.

I'll happily endorse (1). I'll say that I know it to be true. I'll say that I'm certain it's true. And I'll say again that, if there are any moral-conceptual freebies, (1) is probably one of them. But I don't think the certainty that attaches to (1), and ought to attach to it, has much if anything to do with its status as a moral-conceptual freebie. 

Here's another claim that might well be a moral-conceptual freebie:

(2) Only human beings have moral rights.

It may be that speciesism is so deeply ingrained in us and in our society that it has infected our moral concepts, such that our concept of moral rights guarantees that (2) is true, and denial of (2) manifests conceptual incompetence. Nevertheless I do deny (2), because I believe that chickens, pigs, cows, goats, sheep, cats, and dogs all have moral rights.