In CGEL (p. 640), it is stated (without argument) that in the sentence
 They live n̲e̲a̲r̲ ̲h̲e̲r̲e̲.
we have the preposition phrase (PP) near here, which (says CGEL) has the following structure: near is the head (in particular, it is a preposition), and it has here as a PP complement. Let's call near here the 'outer' PP, and here the 'inner' PP. The inner PP (GGEL continues), in turn, is headed by a non-specified preposition (see the next paragraph). In this case, the inner PP consists of just the head, but here is an example where an inner PP has a more complex structure: from u̲n̲d̲e̲r̲ ̲t̲h̲e̲ ̲b̲e̲d̲.
To say that the head of the inner PP is not specified means that instead of here, we could have other things, e.g. there, London, etc. This is in contrast to PP complements headed by specified prepositions, where nothing else could replace that preposition, e.g. of in out of the box.
Now, my question: what is the argument for this proposed structure of near here? Let's assume that near is indeed a preposition (which is not obvious, but maybe it's indeed true; see below). Why not say that here is the head? Is it just that near appears first? Surely there is more to it than that---there are all kinds of cases where the head of a PP is not the first word. Here are some examples (the head is underlined): incredible t̲h̲o̲u̲g̲h̲ it seems; one b̲y̲ one; ten years a̲g̲o̲; these issues a̲s̲i̲d̲e̲ (CGEL, pp. 631-634). Yes, the default position of the head in a PP is the initial position; yes, these examples are all exceptional in one way or another. By how do we know that near here isn't also somehow exceptional? The fuzzy status of near (see below) makes near here a good candidate for something exceptional to happen...
Also, note that either near or here can be omitted from :
 a.They live near.
b. They live here.
Finally, note that semantically at least, either one can be taken as the essential one:
 a. They live near (by which I mean: near to where we are now).
b. They live here (well, not exactly, but almost).
I am tempted to say that [3a] is a rephrasing of  when near is the head of near here, while [3b] is a rephrasing of  when here is the head of near here.
One line of argument, suggested by sumelic, is that near always precedes its complement when that complement is an NP:
 They live n̲e̲a̲r̲ ̲[̲m̲y̲ ̲h̲o̲u̲s̲e̲]̲/̲[̲w̲h̲a̲t̲ ̲c̲o̲u̲l̲d̲ ̲b̲e̲ ̲d̲e̲s̲c̲r̲i̲b̲e̲d̲ ̲a̲s̲ ̲a̲ ̲l̲a̲k̲e̲]̲.
In these cases near cannot be omitted, so it is definitely the head.
Moreover, sumelic points out the following sentence:
 If your hotel is near here or near the train station, you can get there on foot. (source)
I don't think  in and of itself proves much. But the following sentence, if acceptable, perhaps would go some way towards proving that near is indeed the head in near here:
 If your hotel is near here or the train station, you can get there on foot.
It would be hard to believe that the common element, near, could be the head when joined with one coordinate, but not the head when joined with the other.
For what it's worth, to my ear,  is indeed OK.
Is this right? Is  acceptable, and if so, does it prove that near is the head of near here?
The 'fuzzy' status of near
As a separate issue, there is some doubt about whether near is even a preposition in  (and if it is not, then it clearly isn't the head). I am still digesting the very interesting answers that sumelic and tchrist have given as regards this topic, and will not yet attempt to summarize them. My original source of the doubt about whether near is really a preposition was that, after all, the following sentence is acceptable:
They live v̲e̲r̲y̲ ̲n̲e̲a̲r̲ ̲h̲e̲r̲e̲.
(And here are some examples in published literature of such usage.)
This would suggest (I thought) that near in  is not a preposition at all, but rather an adverb modifying here. And if this is correct, then it is here which is the head of near here, contrary to what CGEL says.
(If there is any doubt that CGEL says what I say it says, here is a direct quote, with the near here example in boldface:
The PPs here, there, now, then occur as complement to a wider range of prepositions than do such PPs as under the bed or after six. We find, for example, They live n̲e̲a̲r̲ ̲h̲e̲r̲e̲; Put it o̲n̲ ̲t̲h̲e̲r̲e̲: I found it b̲e̲h̲i̲n̲d̲ ̲̲h̲e̲r̲e̲: You should have told me b̲e̲f̲o̲r̲e̲ ̲n̲o̲w̲; He certainly stayed p̲a̲s̲t̲ ̲t̲h̲e̲n̲.
Note that the other CGEL examples don't suffer from the same problem as near here, because what are claimed to be the heads of the matrix PPs (on, behind, before, past) cannot be preceded by very. Before and past can be preceded by e.g. much and way, but not by very. )
Again, several answers have addressed this issue (whether near is really a preposition) already, and I am still working through them. Moreover, as tchrist and others have pointed out, it has been already discussed in other questions, where this one seems the most relevant.
Assuming we accept that near is a preposition in , what argument can be given for the assertion that it is near, rather than here, that is the head of the PP near here?
In particular, is  acceptable, and if so, does it prove that near is the head of near here?