How well do those Park Tool chain cleaners work on fixed gear?

I remember when I got a chain cleaner ages ago (a Barossi or something) it was a revelation for cleaning the chains on my geared bikes, but it's a pain in the arse to use on a fixed gear. The handle on the Park one makes it look like it could be much better for fixed, but do they do okay with a fixed chain that doesn't have all the slack of a geared bike?