By the way, I understand that you might be more interested in some form of a harder evidence. But I thought it may be useful to you (or somebody out there) to hear the above philosophy which I personally have found quite useful.

I have asked this type of question in several occasions from various people. The most satisfying answer I have received is that of my advisor Madhu Sudan paraphrased here: the main reason it is wise to assume different computational classes (such as levels of PH) are different is that there is no good reason for them being equal. Although coincidences sometimes occur -- especially when one combines computational resources which are not easily comparable -- (here I am hinting at IP=PSPACE) it seems reasonable in normal contexts to trust the general principle of "more resources, more power".

Hmmm I am a bit confused now for the case of $\epsilon=\frac{1}{N}$. It seems this paper arxiv.org/pdf/quant-ph/9605034v1.pdf states that with about $\frac{\pi}{4}\sqrt{N}$ iterations one can get a high probability result, i.e. $\epsilon=\frac{1}{N}$. (page 2 bottom of first column) On the other hand the paper you mentioned says, in page 4 end of section 3, that $o(1)$ failure probability is impossible for $O(\sqrt{N})$ queries.