question about instrument proficiency check

Until recently I was convinced that only a CFII could
conduct an instrument proficiency check (as opposed
to a plain CFI), but now I am confused:

14 CFR 61.195(c) says that an instructor must have an
instrument rating on *both* his pilot and instructor
certificates, i.e., be a CFII, in order to "provide(s)
instrument flight training *for the issuance of an
instrument rating*";

This is how a plain CFI (not II) can provide the training
required to satisfy the requirements of good old
14 CFR 61.109(a)(3) -- i.e., the three hours training
by sole reference to instruments required to get a
private certificate;

Now, the problem is that I haven't found where in the
regulations it is specified that a CFII must conduct
the instrument proficiency check (which is *not* training
for the issuance of an instrument rating that the student
already has); 14 CFR 61.57(d) says training has to be
provided by an 'authorized instructor' (in addition to
examiners etc.) -- is this what I am missing? that
would be an odd use of the term 'authorized instructor'
as compared to other places in the regs.

Now I probably missed something obvious, but can a
CFI (not II) provide the training and endorse someone's
instrument proficiency check? (and if not where is it
said in the regs?)

Note: I am not trying to do something silly, but
I am in the process of studying the regs in details
and I like to understand the fine points.

--Sylvain