I don't use a stop bath between Pyrocat HD and TF-4; so no, I haven't tested it myself. The use of an alkaline fix for Pyro negatives is something I read. I likely read it in one of Sandy King's articles, the Pyrocat HD instructions, one of Steve Ancell's books or the Book of Pyro. I honestly don't remember where I read it. However, 5 - 10 minutes in an acid fix will have a much greater stain impact then 30 seconds in a mildly acidic stop bath.
Do you use an acid Fix with your pyro negatives?
I use a rapid fixer, either Ilford Hypam or Kodak Rapid Fixer (without hardener) or Kodak C-41 fix. They are all weakly acidic - with a pH of about 5.5 or so. Not what I would term an "acid fix" as they are very close to neutral pH.
(Note: I have played with TF-4, TF-5, and the upcoming Super Universal Fix. I never liked the precipitate/sludge aspect of TF-4, but TF-5 and SUF are both precipitate free and I'll take some more consideration of them in the near future.)
What's your definition of a "mildly acidic stop bath" and what is it's pH? I think you'll find that if you are using a rapid-type fixer, the stop bath is more than 100 times more acidic than the fixer is.
Most stops are going to be less than pH 3 when fresh and the pH slowly increases as the acid gets used up. So a used stop bath will be around pH 4 for much of it's usefull life.
That's a much lower pH and much more acidic than any rapid fixer.
The test I did was comparing fresh acetic acid stop bath (so the pH was about 3 as I stated above) with a running water stop bath. As I said in the earlier post, the difference I saw was slight between the two tests (I used PMK as my staining developer).
There was a slight difference in the overall density of the two films - the water bath film did have a slightly higher density, about 0.02 OD than the stop bath. One may wish to claim it was due to the acid stop lowering the stop bath film density, but I suspect it was due to the water bath film continuing development and not being stopped immediately like the stop bath film was. The difference was so slight, I did not pursue it further and continued to use stop bath.
(Other note: I use a buffered stop bath. I make a strongly acidic acetic acid stop bath, and then I add sodium acetate which buffers the stop. If raises the pH up to about pH 5, while increasing the acidity of the stop bath. It makes for a very fast acting stop bath but with a much milder pH. Keep in mind that acidity and pH are not the same thing.)