Okay, here we go. Hope you don’t mind a long read.
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Existence, as it is used in everyday language, primarily means something actually (tangibly) exists. For intangible objects, this is more complex: does the First Amendment exist other than as a piece of paper?
If we are talking about a concept of a potentially existing physical object, there is nothing strange about saying that the concept exists but the actual object doesn’t. I have at this very moment a clear concept of a lifesize gold statue of myself, but I can assure you that the actual statue doesn’t exist, even though it would be entirely possible to create such a statue. Saying that the concept ‘exists’ is however somewhat odd, since existence is primarily used in connection to physical objects.
So we have two kinds of existence, which you should distinguish:
A. existence of objects, and
B. existence of concepts (of objects)
IPU’s lack A, but the concept of IPU has B. At least, that’s how it appears at first analysis.
If you don’t distinguish A and B, and go around applying questions relating to existence A on things having existence B, you are making a category mistake. There you go, a shiny new word for you to use. 
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Now, if you’re ready, we can complicate things.
2.a.
The concept of Santa Claus exists, but does Santa Claus exist? Obviously not as a tangible jolly man with a beard, but in fact he does exist as a social phenomenon. So do we have an intermediate kind of existence? Not wanting to go too deep in this mess, I would leave it at pointing out that the answer depends on what ‘Santa Claus’ is supposed to refer to: a man or a social phenomenon.
2.b.
Do all concepts exist or are there preconditions? This is a tricky subject, which in fact depends on which precise philosophy of mind you adhere to.
Around the time of Spinoza and Locke there was, IIRC, discussion on whether incoherent ideas (= concepts) really existed. It was thought that if you really thought things through, you would only have coherent, non-contradictory concepts in your head, be wise and know all truth. From there on it was a tiny step to saying those incoherent concepts didn’t really exist (in your head), but where dream images, chimaera. In effect they transposed the whole appearance/reality metaphysics from the physical to the mental domain. (sorry for all the jargon)
I’m not going to rehash the whole debate, which is far too complicated to go over here. I’ll just state that I do not see why a contradictory or incoherent concept doesn’t in itself have existence when it is thought, even if it might disappear after close scrutiny. As long as you think about it, it exists.
2.c
Another take is that we rate concepts according to the possibility of their existence. But this is again a difficult road to travel. Time machines or perpetuum mobilea, as far as we know do not and cannot exist. They are, however, very much talked and though about. Cold fusion AFAIK doesn’t exist at the moment, but could in principle very well be possible.
So the concepts of these objects/phenomena are discussed and insofar these concepts do exist, but they have varying degrees of possibility of existence, at least for the scientist. Layman may not know the difference.
Hence the mere possibility of realization of the object that the concept refers to does not seem to be a good way to distinguish between kinds of concepts.
2.d
We can go a step further, though. We have seen that a concept that you are thinking of does have existence while being thought. Could you also speak of the concept having existence in itself, without someone thinking of it? This is more tricky. We are perilously close to Platonic heaven.
With mathematical concepts, people normally do not have much compunction of giving them a kind of existence of ‘stable’ concepts regardless of specific persons thinking about them. With really outrageous ones, like the gold statue I just mentioned, the existence is not much more than the existence of any fleeting image. So we seem to think implicitly in grades of existence, when applying this term to concepts.
Now how would the IPU rate? The problem with the concept is, and that may be the heart of your real problem, that it appears to be internally incoherent, hence the concept has a structural problem. The attribute ‘color’ is usually meant to refer to the actual color of something. ‘Invisible’, however, by definition refers to something not having any kind of visibility, which precludes it having color. So how can an object be both pink and invisible? The concept appears to be inherently flawed and for that reason, you could say, negates itself, doesnt’ ‘exist’. (I’m bypassing the problem of finding a unicorn in the first place)
In the OP you refer to a unicorn which is first made pink, then turned invisible. Wouldn’t that be an invisible pink unicorn? It depends. You are now using pink in a different sense than the common one. ‘Pink’ is usually refered to the actual color of something, not the color it might have if something changed with the object. IIRC most cars are first painted in grey or pink ground paint (base paint?), then sprayed over in their actual color. You surely wouldn’t call every car ‘pink’ for that reason? If we go by your definition, there indeed is not a problem with the idea of an invisible pink something. But that is not the common usage of the attribute ‘pink’.
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Summarizing all this:
- IPU’s as real objects don’t exist as far as we know.
- IPU’s as concepts have the kind of existence we attribute to all mental images.
- The concept of an IPU is internally incoherent, but I do not find that sufficient reason to deny them all existence. To do that you have to take recourse to a specific philosophy of mind, which must be defended on its own ground.
Whew. Do you now see why I didn’t answer straight away?