Recall that Grothendieck's Homotopy Hypothesis states that the homotopy category of weak globular $n$-groupoids (as in https://arxiv.org/pdf/1009.2331.pdf) is equivalent to the category of homotopy $n$-types.
It's also well known that strict globular groupoids are not enough since $\Pi_3 (S^2)$ cannot be rectified to a strict $3$-groupoid due to the non-vanishing of the Whitehead bracket.
On the other side, it's well known that every quasi-category can be rectified into a strict $(\infty, 1)$-category. More generally, there's Berger-Moerdjik result that says that a bunch of algebras over operads can be rectified. In particular, $A_{\infty}$-algebras can be rectified and also homotopy coherent diagrams (Vogt's theorem).
In view of these observations, I have the following questions:
By considering $\Pi_3 (S^2)$ as a quasi-category (i.e., extend it by degenerated simplices), we can rectify it. Why doesn't it contradict my first observation?
Why can't we rectify weak globular groupoids? By Vogt's theorem, homotopy coherent diagrams can be rectified. So, what fails if one views a weak globular groupoid as a homotopy coherent diagram and, then, rectify it?
Further comments about the intuition of why one cannot rectify weak globular groupoids would be also of great utility.
